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AN INTRODUCTION - -^ 



TO 



ALGEBRA; 

WITH , 

NOTES AND OBSEBYATIOXS : 

DESIGNED 

For the Use of Schools and Places of Public £ducaUou. 
BY JOHJ^" BOjsrj\rYCASTLE, 

OF THE ROTAT. MILITAKY ACADKMY) WOOLWICH. 



Ing^nuas didicisse fideliter artes 

EmolUt mores> nee finit esse feros. 

Ovid. 



THE SECOND AMERICAN EDITION, EEVISJ^B ANP 

CORRECTED. 



FHIIiABEIiPHIA ; 

PXnUJSHED BT KIMBER AM> CONRAD, 
NO. 93, MARKET STREj^T.^ 
* T. d* G. Fahorrt printen* 

1811. 



!'C 






39335^ 






(l. s.) 



DISTRICT OP PENNSYLVANIA, to wit: 
Bs IT REMEMBERED, That on the eleventh day o 
bruaiy, in the thirty .fifth year of the Independer 
the United States of America, A. D. 1811, 
KiMBBR ANO Conrad, 
of the said cUstrict, have deposited in this office the title of a 
the right whereof they claim as proprietors, in the words foil o 
to wit : 

An Introduction to Algebra ; with Notes and Observations 
signed for the Use of Schools and Places of Public Educ 
By John Bonnycastle, of the Royal Military Academy, ^ 
wich. 

— Ingenuas didicisse fideliter artes 
Emollit mores, ncc finit ecse feros. 

Ovid. 
The second American edition, revised and corrected*. 

In conformity to the act of the congress of the United State 
tied, ** An act for the encouragement of learning, by secur 
copies of maps, chatts, and books, to the authors and pre 
oi such copies during the times therein mentioned." An» 
the act, entitled, ** An act supplementary to an act, entitle 
act for the encouragement of learning, by securing the r 
maps, charts, and books, to the authors and proprietors 
copies during the times therein mentioned," and exter 
benefits thereof to the arts of designing, engraving, and et 
torioal and other prints.** 

D. CALDWEI 
Clerk of the District of Pair 









THE acknowledged merits of the folloxving 
work J and its peculiar fitness for the use of ma- 
thematical students^ has induced the Editor to 
give it a careful revision: the errors of former 
editions and ^me improprieties in the original 
'e corrected and removed. 

E. L. 

New-Garden Boarding-school, 
U mo. 1810. 



A 

4 



PREFACE. 



THE powers of the mind, like those of the body, are 
increased by frequent exertion ; application and industry 
««ipply the place of genius and invention ; and even the 
creative faculty itself may be strengthened and improved 
by use and perseverance. Uncultivated nature is uni- 
formly rude and imbecile^ it being by imitation alone 
that we at first acquire knowledge, and the means x)f 
extending its 1x)unds. A just and perfect acquaintance 
with the simple elements of science, is a necesBary steg 
towards our future progress and advancement : and this>. 
assisted by laborious investigation and habitual inquiry, 
will constantly lead to eminence and perfection. 

Books of rudiments, therefore, concisely written, welF 
digested, and methodically arranged, are treasures of 
inestimable value; and too many attempts cannot be 
made to render them perfect and complete. When 
the fil*3t principles of any art or science are firmly fixed 
and rooted in- the mind, their application soon becomes 

a2 



▼i PREFACE. 

easy, pleasant, and obvious ; the understanding is delight- 
ed and enlarged ; we conceive clearly, reason distinctly, 
and form just and satisfactory conclusions. But^ on the 
contrary, when the mind, instead of reposing on the sta- 
bility of truth and received principles, is wandering in 
doubt and uncertainty, our ideas will necessarily be con- 
fused and obscure ; and every step we take must be at- 
tended with fresh difficulties and endless perplexity. 

Tjiat the grounds, or fundamental parts, of every 
science are dull and unentertaining, is a complaint uni' 
versally made, and a truth not be denied ;f but, then, 
what is obtained with difficulty is usually remembered 
with ease ; and what is purchased with pain is possessed 
\i4th pleasure. The seeds of knowledge are sown in 
every soil, but it is by proper culture alone that they are 
cherished and brought to maturity. A few years of early 
and assiduous application never fails to procure us the 
reward of our industry ; and who is there, who knows 
the pleasures and advantages which the sciences afford, 
that would think his time mispent, or his labours use« 
less ? Riches and honours are the gifts of fortune, ca- 
sually bestowed, or hereditarily received, and arj fre- 
quently abused by their possessors ; but flie superiority 
of wisdom and knowledge is a pre-eminence of merit 
which originates with the man, and is tlie noblest of all 
distinctions. 

Nature, bountiful and wise in all things, has provided 
us with an infinite variety of scenes, both for our instruc- 
tion and entertainment; and^ like a kind and indulgent 



PREFACE. vii 

parent} admits all her children to an equal participation 
of her blessing^. But^ as the modes, situations, and cir- 
cumstances of life are various, so accident, habit, and 
education have each their predominating^ infiuence, and 
give to every mind its particular bias. Where examples 
of excellence are wanting, the attempts to attain it are 
but feW'; but eminence excites attention, and produces 
imitation. To raise the curiosity, and to awaken the 
listless and dormant powers of younger minds, we have 
only to point out to them a valuable acquisition, and the 
means of obtaining it. The active principles are imme- 
diately put into motion, and the certainty of the conquest 
is ensured from a determination to conquer. 

B«t, of all the sciences which serve to call forth this 
spirit of enterprise and inquiry, there are none more 
eminently useful than the mathematics. By an early 
attachment to these elegant and sublime studies, we ac- 
quire a habit of reasoning, and an elevation of thought, 
which fixes the mind, and prepares it for every other 
pursuit. From a few simple axioms, and evident prin- 
ciples, we proceed gradually to the most general propo- 
sitions and remote analogies: deducing one truth f^m 
another, in a chain of arginnent well connected and lo- 
gically pursued; which brings hs, at last, in the most 
satisfactory manner, to the conclusion, and serves as a 
general direction in all our inquiries after truth. 

And it is not only in this respect that mathematical 
learning is so highly valuable; it is, likewise, equally 
estimable for its practical utility. Almost all the works 



• • • 



t»i PREFACE. 

of art) and devices of man, have a dependence up 
principleSf and are indebted to it for their origii 
perfection. The cultivation of these admirable sci 
is, therefore, a thing of the utmost importance 
ought to be considered as a principal part of every 
fal and v^ell-regulated plan of education. They ar 
guide of our youth, the perfection of our reason^ ar 
foundation of every great and noble undertaking. 

From these considerations I have been induc( 
undertake an introductory course of mathematical sci 
and, from the kind encouragement v^hich I have hit 
i*eceived, am not writhout hopes of a continuance c 
same candour and approbation. Considerable pn 
as a teacher, and a long attention to the difHcultie! 
obstructions which retard the progress of fearne 
general, have enabled me to accommodate mysel 
ipore easily to their capacities, and understandings. 
as an earnest desire of promoting and diffusing i: 
knowledge is the chief motive for this undertaking, i 
pains or attention shall be wanting to make It as com 
and perfect as possible. 

The subject of the present performance is A 
BRA^ which is one of the most important and u 
branches of those sciences, and may be justly consic 
as the key to all the rest. Geometry delights u 
the simplicity of its principles, and the elegance c 
demonstrations. Arithmetic is confined in its ob 
and partial in its application: but algebra, or 
analytic art, is general and comprehensive, and 



PREFACE. ix 

be appUed with success in all cases where truth is to be 
obtained and proper data can be established. 

To trace this science to its birth, and to point out the 
various alterations and improvements it has undergone in 
its progress, would far exceed the limits of a preface. It 
wiU be sufficient to observe, that the invention is of the 
highest antiquity, and has challenged the praise and admi- 
ration of all ag^s. Diofihantus appears to have been the 
first, among the ancients, who applied it to the solution of 
indeterminate or unlihiited problems ; but it is to the mo* 
demsthat we are principally indebted for the mosfcurious 
i^ement^ of the art, and its great and extensive useful- 
ness in every abstruse and difficult inquiry. JW w/on, Ma* 
clauririj Saunderson, Simfiaony and Emeraotij are those, of 
oor own countrymen, who have particullarly excelled in 
^8 respect ; and it is to their works that I would refer the 
young student, as the patterns of elegance and perfection. 
The following compendium is formed entirely upon the 
iiiodel of those writers, and is intended as a useful and ne- 
cessary introduction to them. Ahnost every subject, 
^hich belongs to pure algebra, is concisely and distinctly 
^tedof; and no pains have been spared to make the 
^hole as easy and intelligible as possible. A great num- 
^r of elementary books have already been written upon 
^s subject ; but there are none, which I have yet seen, 
but what appear to me to be extremely defective. Besides 
being totally unfit for the purpose of teaching, they are 
generally calculated to vitiate the taste, and mislead the 



X PREFACE. 

judgment A tedious and inelegant method prevails 
through the whole, so that the beauty of the science is 
generally destroyed by the clumsy and aukward manner 
in which it is treated ; and the learner, when he is after- 
wards introduced to some of our best writers, is obliged 
to unlearn and forget every thing which he has been at so 
much pains in acquiring. 

It is in the sciences as in every branch of polite litera- 
ture ; there is a certain taste and elegance which is only 
to be obtained from the best authors, and a judicious use 
of their instructions. To direct the student in his choice 
of books, and to prepare him properly for the advantages 
he may receive from them, is, therefore, the business of 
every writer who engages in the humble, but useful task 
of a preliminary tutor. This information I have been care- 
ful to give, in every part qf the present performance, 
where it appeared to be in the least necessary ; and, 
though the nature and confined limits of my plan admit- 
ted not of diffuse observations, or a formal enumeration 
of particulars, it is presumed nothing of real use and im- 
portance has been omitted. My principal object was to 
consult the ease, satisfaction, and accommodation of the 
learner ; and the favourable reception the work has met 
with from the public, has induced me to give this edition 
an attentive and careful revisal. 



THE CONTENTS. 



Page 
Definitions - - - - ^ . . 13 

Ad^lion of algebra 16 

Subtraction of algebra - - • - , 20 

Multiplication of algebra - - - - <> 3 1 

Division'of algebra ^ - • - - - 24 

Algebraic fractioni^ - - - - - 28 

Involution, or the raising of powers • - - 43 

Evolution, or the extraction of roots - ^-v 47 

Of irrational quantities^ or surds - - - 52 

Of infinite series - - . - . - 63 

Of arithmetical proportion - - - - 70 

Of geometrical proportion ... " 73t 

Of simple equations - - - - - 76" 

Of quadratic equations 99 

Of the nature and formation of equations in general 113 

Of the resolution of equations by various methods 423 

Of the resolution of cubic equations - - - 128 

Of the resolution of biquadratic equations - ISl 

To find the roots of equations by approximation and 

converging series 1^6 

To find the roots of pure powers in numbers - 1 41 

To find the roots of exponential equations - - 14^ 

Of indeterminate or unlimited problems - - 145 

Of diophantine problems - - - - 152 

Of the summation and interpolation of infinite series 1 63 

Of logarithms 190 

A collection of miscellaneous questions - - 21^ 



ALGEBRA. 



DEFINITIONS- 



ALGEBRA is the art of computing by symbols. 

1. L.ike quantitiea are those which consist of the same 
letters. 

2. Unlike quantities are those which consist of diffe- 
rent letters. 

3. Given quantitiea are those whose values are 
known. 

4. Unknown quantities are those whose values are un- 
known- 

5. Simfile quantities are those which consist of one 
term only. 

6. Comfiound quantities are those which consist of se- 
veral terms. 

7. Positive or affirmative quantities are those which 
are to be added. 

8. Negative quantities are those which are to be sub- 
tracted. 

9. Like signs are all affirmative (-f)j or all negative 

(-)• 

10. Unlike signs are when some are affirmative (4-^ 

and others negative ( — ). 

11. The co»efficient of any quantity is the number pre- 
fixed to it. 

B 



14 EXPLANATION OF 

12. A binomial quantity is one consisting of two 
terms; a trinomial of Uiree terms; a quadrinomial of 
four, &c. 

13. A residual quantity is a binomial where one of 
the terms is negative. 

i4. llic flower of a quantity is its square, cube, bi- 
quadrate, &c. 

15. The index or ex/ionent of a quantity is the number 
•which denotes its root or power. 

16. A surd or irrational i/uaniity is that which has no 
exact root. 

17. A rational quantity is that which has no radical 
sign (v/) or index annexed to it. 

.18. The recifirocal o[ dA^y quantity is that quantity in- 
verted, or unity divided by it. 



EXPLANATION OF THE CHARACTERS. 

+ Is the sign of addition. 

— _ of subtraction. 

X . of muhiplicalion. 

-5- — of division. 

: : : : of proportion. 

y/ — . of the square root. 

^ — of the cube root. 

s= — of equality. 

Thus, a 4" ^ sliows that the number represented by b 
is to be added to tliat represented by a. 

a— 6 shows that the number represented by b is to be 
subtracted from that, represented by a. 

aTjb reprtsents the difi'crence of a and b when it is 
not known which is the ij^reatest. 

ab^ OT axb^ or a.b denotes the product of the numbers 
represented by a and b. 



tml: characters. 15 

a 
a-T-by or — , shows that the nunib's^r represented by a 

is lobe divided by that reprcsciUed by /;. 

a \ b I : c I d denotes iiiat a is u\ the same proportion 
to A as c is to d, 

jc = u — b + c is un e(iuutioii, siiowing that x is equal 

to the differ*, nee of a and b^ ud\)cd lo the quantity c, 
1 _i 

V^a, or a*', is the square root of :; ; ^/a^ or a^, is the 

cube root of a; and u" is the mU root of r?. 

c2 is the square of a; a^ u.^ cube of a\ w^ the fourth 
power of a ; and a"* the //nil power of a, 

— is the reciprocal of — , and — the recipi*ocal of a. 



a 



a 4- 6 X c» or (a + b)c is the product of the compound 

quantity a + b multiphed by the simple quantity c, 

■ - d ^ I) 

a + 6 -7-a — 6, or (a + b)-^{a — b)^ or , is the 

quotient of a + ^ divided by a — b. 

\/ ab -^ ^d or {ab + cdy is the square root of the com- 
pound qn-intity ab -p cd, 

a -f- 6 — c^ or (a + A — c)^ is the cube, or third power, 
of the quantity a-^- b — c. 

5a denotes that the quantity a is to be taken 5 times, 
and t{Jb -f c) is 7 limes b -f- c. 

It is also to be remarked that the sig-n + is generally 
expressed by the word filuH^ or more^ and the sign — by 
minus J or le/iS. 

And, in the computation of problems, it must be ob- 
served, that the first letters ol ihe- alphabet are usually put 
for known quaniilies, and liie hist lor those which ar*j 
unknown. 



16 





ADDITION. 




EXAMPLES 


• 


-*-3a 


+ Sax^ 


+ 60734. 8y 


+7a 


+ 7flx2 


— 3x3+ 7j^ 


+8a 


— 3ax« 


— 13x3+ 8z/ 


— a 


— 4ax^ 


+ 2x3— 3z/ 


—2a 


+ Aax^ 


+ x3— y 


+9a 


+ \ 2ax2 


— 7x3+ 1 9y 


— 2fl2 


+ 8A2y3 


— 3ad+ r 


— 3tt2 


+ 6*2^^3 


+ 3ad— 10 


— . 8^ 


10^2^3 


+ 3aA — 6 


+ 10a2 


20^2^3 


— ah^ 2 


+ I3a2 


— b^y^ 


— ^2ad+ 1 i 


— 5ary 


— 8x2y2 


— 12x2— 8x 


— 3xy 


+ 3jr2i/2 


+ 10x2_3x 


+8xy 


—2x2^/2 


— \4x^+7x 


+7a:y 


+4ar2j/2 


+ 8x2+ :x; 


-^^axi 


— 6\/ax 


~2y+2axi 


-f- flxi 


+ 2y/ax 


+ y+ axi 


— -Sflxi 


— ^ t'^ax 


— 7y— -Saxi 


+7axi 


+ 10-v/ax 


+5y+3ax* 









ADDITION. 



19 



CASE III. 



To add quantities tohich are unlike^ and have unlike tigns. 



y 



RULE. 



Collect all the like quantities together by the last rule^. 
and set down those that are unlike, one aftec another, 
with their proper signs. 



5xy 
4ax 

— xy 
— 4ax 


EXAMPLES. 

2xy — 1 0x2 2ax — 1 50 +2x^ 
— 3x2 ^a:y 3x2+2flrx+6x2 

— 8x2 — j^y ^xy — 3x*+50 
— xy +9x2 ^x+\QO — 5x2 


4xy 


xt/ — 12x2 


4ax+4x2+4xy 


6x»y2 
— 4x2y 
— ^2<2xy • 
— 3x2y 


12xfl — x2 

4«:f+xy 

3y2— tfX 

2x2—24 


6+20^ax — 3y 

x+ 4^xy+3y 

y — 2v^ax^— 3y 

20+ Zs/ax^^3y 






.* 


3x2y 
— 3xy« 
— 3y2x 
— 8x2y 


2v^x — 8y 
3v^xi/+IOx 
2x+>x+y 
—8 + y/xy 


• 

a^— 8+x*— 2 
a — lQ+a2 — jc 

X*— a* + 8~4 
10— a— x2— y 







[ 20 ] 



SUBTRACTION. 



RULE. 



Change the signs of all the quantities to be subtractec 
or conceive them to be changed, and then collect th 
different terms together as in addition. 



5a2 — 25 
2a2 — 5b 



EXAMPLES. 



6jc2-^ 8v+3 Sixy — 2+ 8jc — y 
2jr2+ 9^/— 2 3xy — 8 — 8^ — 3y 



3a2+3d 4jc*— .l7i/+5 2arz/+6-f 16x+2z/ 



Zxy — 8 ^y^—y — 1 ■ — 10 — 8x — 3xz/ 

— xy+B y^+y+1 xy — 7x+Z — Aioy 



4xy — 16 t/2 — 2z/— 2 ■ — 13 — jC — 4xy-^4ay 



Sx^y — 8 4^xy^--Xy/xy 
.3x2j/+l 2^xy+2 + xy 

^ V 



5'ar2-f ^a-— 8— 4^* 
6x2—10+43 — x^ 



oxy — 20 4x^ — 3.(fl+6) xy^ + \Oax/(xy+ 10) 
4xy — 30 3f3 — 8.(a-f^) ^2^2+ 2a^\xy-^lO) 



E 21 ] 

MULTIPLICATION. 

CASE I. 
When both the factor a are timfile quantities* 

RULE. 

Multiply the co-efficients of the two terms together, 
4nd to the product affix ail the letters in those terms, 
aud the result will be the whole product required. 

Xote*, Like signs produce +, and unlike signs — . 



12a 
3d 


EXAMPLES. 

—2a Sa 

+4b — 6x 


— 9x 
—56 , 


36ad 


— Sab - 


— 30ax 


+,4,5bx • 


7ab 
-^Sac 


6a^x 

Sx ' 


_jc2y 

xy^ 


— ^y 


— SSa^bc 


30o2x» 


— x^y^ 


+7x^y^ 


— 5ax 
3x 




+5xy 
—3 


— 7xyz 
— 6a:c" 








\ 


8ax2 
Sax 


Sx^y^ 
x^y^ 


— .3j:i/2 


\7x^y^> 
2ax^ 







* That like signs make +, and unlike signs — , in the product, 
may be shown thus : 
Xst. When + a is to be multplied by + 5 : this implies that -f" " 



22 



MULTIPLICATION. 



CASE IL 

When one of the factors is a corn/iound c^uajitity ^ 

RULE. 

Find the products of the multiplier, and every terrr ^ 
the multiplicand, separately, and place them one tv ^ 
another, with the proper signs, and the, result will be 
whole product required. 



4(2 — 2 b 
Sa 



EXAMPLRS. 



6a:y — 8 
2x 



a2 — 2x+6 
xy 



12a2 — 6ab \2x^y^^\6x u^xy — 2x^y+&xy 



13jc— ad 
12a 



S5j>— 7a 



•X 



3y_8+2xv 
xy 



2x^+ay 
2xy 



12jc2 4y2 

— 2.r2 



2t/2 — Sx^-^r.v 
Zxi/^ 



is to be taken as manv times as there are units in b-y and, since 
sum of any number of affirmative terms is affirmative, it is plain 
{+a)X{+b)=-\-ab. 

2. If two quantities are to be multiplied together, the i 
wi]\ be exactly the sa-ne, in whatever order they are pl- 
a times b is the same as b times a; aiid, the 



for 



\ 



• 



MULTIPLICATION. - 23 

CASE III. 

ken both the factors are comfiound quaiitities, 

RULE. 

iply every term of the multiplier into every term 
nultiplicand, respectively, and set down the pro- 
le after another with their proper signs, and their 
11 be the whole product required. 





5x+4>y 
3x — Sy 


x2-f.jiryJi-»z/2 

X — t/ 


+ J,.2 


\5x^+\2xy 


x^-^x^y — xy^ 
— x^y — xy^-j-y^ 


?/+?/% 


\5x^ + 2xy — 8 1/2 


x^ * — 2xy2^-y^' 




x^+y 
x^-^y 


x^+xy+y^ 
X — y 


7/2 


x*+yx^ 

^^yx^+y^ 
r 


x^+x^y-^xy2 
.•^x^y'-^xy^ — 1/3 


— 1/2 


a7*4.2i/x2-f-!/2 


X"* * * — y^ 









a is to be multiplied by -{-b, or +6 by — a, this is the 
taking — <z as many times as there are units in -j-^; and 

sum of any number of negative terms is negative, it is 
hat (— a)x(-f-^) or (-j- 6)x(— fl)=— a^- 
hen — a is to be muhiplied by — b-. here a — a:=o; 
! (fl-r-fl)X — ^ Js also ^o, because o multiplied by any 

produces o; and since the first term of the product, or* 
), is, by case 2, = — ab^ the last term, or (— a)x(— ^), 
rss-\-ab, in order to make the sum ( — ab^ab)wmo't con- 






24 • MULTIPLICATION. 



EXAMPLES FOR PRACTICE. 

1. Multiply 12ar by 3a.' Aub, 36a2jt 

2. Multiply 4x2 — 2y by 2y. ^w«. Sx^y— 4i/' 

3. Multiply 2A^+4y by 2j:-.-4j/. -in«. 4a;^ — I6y' 

4. Multiply x2 — ^J7v4-y2 by jc+y. Ana. x^+y^ 

5. Multiply x^+x^y-^xy^+y^ by jc — y. Ana. x* — y^ 

6. Multiply x^+xy+y^ by jr2 — ^y+y^, 

r. Multiply 3a:2 — ^xy+5 by jr^^gjri/ — 3. 

8. Multiply 2a2-— 3qx+4x2 by sa^ — eax — 2x^, 

9. Multiply 3x3+2jr2i/2+3i/3'by 2x3 — 3x^y^+5y^, 



DIVISION. 

CASE L 
IVhen the divisor is a simfile gtcantiiy, 

RULE. 

1. Place the dividend above a small right line, and the 
divisor under it, in the manner of a vulgar fraction..' 

2. Expunge those letters which are common to both, 
the dividend and divisor, and divide the co-efficients of all 
the terms by any number that will divide them without ^ 
remainder, and the result will be the quotient r^qumd. 

A'ote*. Like signs make +, and unlike signs*— ^ the 
same as in multiplication. 

. ■ ^ -I KJ 

* That like signs give -{-, and unlike signs — , in tixt qnoti 
will appear thus : 

The divisor, multiplied by the quotient, most produce the dividend; 
therefore. 



DIVISION. S5 

EXAMPLES. 

I. It is required to find the quotient of a-f-a ; 8dc-^2^; 

a&d adc-^dcd. 

. a , 86c ^ - abc a 

^««. — .==1; _=s4c; and -7^-. = —. 
a 2o bed d 

3. It is required to find the quotient of l^xy-^^x^^ 

and (ad+da)-i-2A. 

3. Divide }Sx^ by 9jr. ^w«. 24:. 

4. Divide lOx^j/^ by — 5x^y. Jna. — 2y. 
5« Divide — 9tf:r2[/» by Qx^y. ^n«. — «y. 
6. Divide — Bjc^ by — 2x. Jm, +4jc.. 
7.' Divide 10a6+15flc by 5a. Ans. 26+3c. 

8. Divide 30ax — 54x by 6x. ^n«. Sa^fff. 

9. Divide lOx^z/ — l5y* — Sy by Sy. 

jins^ 2^7*— 3y— I. 
% . 10. Divide \^a+Zax—\7x^ by 2 la. 
11. Divide 3a«— 15+6a+3A by So. 

CASE 11. 

When the divisor and dividend are both com/iound 

quantities* f 

. Ji* RULE. 

' h Range the terms of both the quantities accord- 
^ to the dimensions of some letter m them, so that 



1 Vfifn both the.terms are -|«, the quotient is -{-, because (-{-) 
X(^ pfodac ci -|- in the dividend. 

they are both — , the qqotient is als» +» because 
(^||}((^} produces — in the dividend. 

one of them is + and the' other «-, the quotient is --^ 
{mJ)x(^) produces — in the dividend; and (— }x(— > 
•1. in the dividend. 

C 



i 

k 



^6 DIVISION. 



the first term may contain the highest power ol 
letter, the second term, the next highest power ; a 
on. 

2. Divide the first term of the dividend h] 
first term of the divisor, and place the result in the 
tient. 

3. Multiply the whole divisor by the term thus f 
and subtract the result from tlie dividend. 

4. To this remainder bring down as many ten 
the dividend as are requisite for the next operatior 
divide as before ; and so on, as in common arithmet 

J^ote, If the divisor be not exactly contained i 
dividend, the quantity which remains after the opei 
is finished must be placed over the divisor, like a v 
fraction, and set down at the end of the quotient, 
coinmon arithmetic. 

EXAMPLES. 

J? -f- y)a:2 -[- 2x1/ -}- t/2(x-|- y 



xy+y^ 
xy+y^ 



a+x)a^+Sa^x+Sax^+x\(fi+4ax+gc'^ 
a'+ a^x 



4a*x+4ax^ 



ax^+x^ 
ax^-i-x^ 



4 



DIVISION. 23 



X — S)j:3—9:c8+2r;r— 27(0:2— 6ar+9 
x^ — 3x2 



Sx^+27x 
-6x2+18jc 



9x — 27 
9jr_27 









ax^'-^x^ 



^^y)34— 1/4(^3 + d2y 4- dy2 + y 3 
64_63y 






b^y^ — y4 
^21/8 — by^ 



by^—y^ 
by3^y* 






2S FRACTIONS. 



EXAMPLES FOR PRACTICE. 



1. Divide a^+2ax+x^hya+a:, Ana.a+x, 

2. Divide a^ — 3a^y+3ay^ — y^ by a — y. 

jin9* a^^^2ay'\-y^. 

3. Divide 1 by 1 — x. Ana, X+x+x'^+x^^^c. 

4. Divide ^x^ — 96 by 3x — 6. 

Ana. 2x3+4j72-|-8a:+16, 

5. Divide a* — Sa^x+XOa^x^-^XOa'^x^+Sax^ — x« 

by a« — 2ax+xK Ana. a^ — 3a«j:+3ajr«— x^. 

6. Divide 48x3 — 760^2 — 64a2x+ XOba^ by 2x — 3«. 

7. Divide y<* — Sj/^x^+Sy^x^ — x« by y3 — Zy^x+Zyxf^ 

—x\ 



ALGEBRAIC FRACTIONS. 

PROBLEM I. 

To reduce a mixed quantity to an imfirofier fraction, 

RULE. 

Multiply the integer by^ the denominator of the ihlic- 
tioHyland to the product add the nuinerator; and the 
denominator being placed under this sum, will give the 
improper fraction required. 

BXAIIPLES. 

5 b 

1. Let 3^9 and a-«<- be reduced to improper frac* 

tions. 

«5 3x7+5 21+5 26 ^ 

Firaty 3—. = --U. c= — ^ =x — , Ana, 

7 7 7 7 

b aXC'^b ac^^b 
And a—— ss ' ■ ■ ss ■ > Ana, 

C € C 



.w 



FRACTIONS. ^ 

3. Let a? H and op-— be reduced to impFo 

ax ' 

|cr fractions. 

^ . x^ xxa+ar3 ax-\'X^ ^ 

WiTBtyX H as =s , the Answer, 

a a a 

^no a:— = = ^ ^n*. 

JT a: or 

• « ^ 6 . ^ . .62 

3. Reduce 8 — to an improper fraction. Ana. — 

7 7 

fix Q ^JC 

4. Reduce 1— — to an improper fraction. Ana, 

a •^ *" a 

5. Let J?— — be reduced to an improper frac 

tion. 

2x— -8 

6. Let 10 H ;: be reduced to an improper frac 

t 

lion. 

7. Let a + r be reduced to an impropei 

o 

fraction. 

X — 3 

• •. Let 1 + 2ar— . be reduced to an impropei 

5x 



fl'action. 



PROBLEM n. 



To reduce an imfirofier fraction to a whole or mixec 

itity. 




RULE. 



Divide the numerator by the denominator, for the 
integral part, and place the remainder, if any, ovei 
the denominator^ and it will be the mixed quantity 
required. 

C2 



Sff PRACTIOS®. 



EXlMPt.ES. 

17 cw 4- a^ 

1. Let -»- and ♦ ' ■■ » be reduced to whole or miKw 

5 X 

quantities. 

If 
firsts — =s= l7-T-5aB3f, ^/%^ cmawer required, 
5 

ax4-a^ a* 

X ' X 

ab'-^a'^ ay 4- 2y^ 

2. Let — - — and — be reduced to whole o 

mixed quantities. 

d o * 

35 Sad — ^ 

3. Let -^ and ' •■ ■ ' " ■ be reduced to whole or mrxe 

8 a 2 ^2 

quantities. Jna. 4— and Sb 2. 

8 a 

4. Let -— - and -— — be reduced to whole or mixe 

zx ' a— >:r 

quantities. 

jf2 y« jj-S y^ . 

5. Let J—— and — — be reduced to whole c 

mixed quantities. 

1 Ox^ •— ^ X -I- 3 

6. Let - — ■ ' ' ' • be reduced to a whole or mixe 

5x 

quantity. 

I2x3+3x* 
T. Let i j g -' ' ' ^1^ - ' be reduced to a whole o 

mixed quaotitjr^ 



. * 



FRACTIONS. SI 

PROBLEM IIL 

To reduce Jractiona of different denominatorsi Po others qf 
the same value^ vfhich Tthall have a common denominator, 

RULE. 

Multiply every numerator, separately, into all the 
denominators but its own, for the new numerators, and 
ail the denominators together for the common denomi- 
oator*. 

• EXAMPLES. ^ 

a h 

I. Reduce—^ and — to fractions of equal values 
be 

that shall have a common denominator. 



or 



xei 



-Of 



6xrasdr 

ac b^ 

-— and — s=/raction$ required. 

be be 

. 2. Reduce -r-, — , and --- ^^ equivalent fractions^ 
be d 

having a common denominator^ 

axcxd:xsacd 

bxdXd=:b^d 

cxpxc^c^b 

bXcXd^bcd 

acd b^d ,cH ^ , , , 

r-v. r-ii and r— , fractions required, 
bed bed bed 

j*Whcft the denominators have a common divisor, it will be 
•^» instead of multiplying by the whole denominators, to multir 
W only by thefe parts which arise from divi(Uii|^tSLi^^ common 



/ 



3S FRACTIONS. 

2x b 

3. Reduce — and — to equivalent fractions, havin 

^ , ^ 2cx a 

a common denominator. Ana, — and— 

ac a 

a a-\-b 

4. Reduce — and to fractions, having a con 

^ ac ab-^i 

mon denominator. Ana, -- and — 7- 

. be be 

3jc 2d 

5. Reduce — , — 5 and ^to fractions, having a con 

. 2* ^^ ^ 9cx 4ab ^ 6aa 

mon denommator. Ana. — — , - — , and --— 

6ac 6ac 6a«. 

3 ^jc 2^ 

6. Reduce -, — , and cH , to fractions, havin 

4 3 a . 

a common denominator. 

9a Sax , 1202+ 04 j 

Ana. — , — —-, and -— 

12a 12a 12a 

7. Reduce ---> — ' ^^^ — ; — > to fractions havin 

2 3 a-^-x 

a common denom nator. 

hi* ri 

8. Reduce — , — , and — , ^o equivalent fraction; 

2a3 2a a ^ 

having a common denominator. 

PROBLEM IV. 

To Jind the greateat common measure of a fraction, 

RULE*. 

1. Range the quantities according to the dimension 
of some letters, as is shown in division. 



* The simple divisors, in this rule, may be easily- found, 
inspection. 



FRACTIONS. 33 

3. Divide the greater term b^ the less, and the last 
divisor by the last remainder, and so on till nothing re- 
nuuDs ; and the divisor last used will be the common 
measure required. 

Mote, All the letters or figures which are common 
to each term of the divisors, must be thrown out of them 
before they are used in the operation. 

EXAMPLES. 

1. To find the greatest common measure of 
ex+x* 

or c+x )ca^+a^x(a^ 



There/ore the greatest common measure is c+x* 
3. To find the greatest common me^\ire of 



x*+%bx+b^)x^ — b^x(x 

x^+2bx^+6^x 



—2dx»~2b^x)x^+ 2bx+b^ 
* or X + b )x^+2bx+b\x+b 



bx+b^ 
bx+b^ 









A 



34 FRACTIONS. 

Therefore x-\-b ia the greateat common aiviaor. 

J 

3. To find the greatest common divisor ©f '. 
jr« — 1 

xy+y 

4* To find the greatest common divisor of 



5. To find the greatest common measure of 

a^b+2a^b^+2ab^+b^' 

PROBLEM V. 

To reduce a /ration to its lowest terms^ 

RULE. 

1. Find the greatest common measure^ as in the last;^ 
problem. 

2. Divide both the terms of the fraction by tLe CQm-| 
mon measure thus found, and it will reduce it to its j 
iDwest terms as was required. 

EXAMPLES. 
CX'A'X^ 

1. Reduce ^ . ^ to its lowest terms. 

ca^+a^x 

ex + x^ya^ + a^x 
fir c-f-a: )ca^+u^x{a^ 
ca^^a^x 



Here, cx-^-x^ ia divided by jr, which ia common to eac 
ferm. 



FRACTIONS. 
Therefore i + «'' «* ^^^ greatest common measur 
and c+jtV r — a=-r ^fraction reguiret 

3. Having -— — - given, it is required to r 

it to its lowest ter ns 



:t'2-|- bx 



bx-\-b^ 
bx + b2 



Therefore x+A is i he greatest common measui 

X^ ■— i A2 J JC^ __ ff -jQ 

3. Reduce^ —--to its lowest terms, jins* — 

x^'^-b^xf 

jr2«_7^2 

4. Reduce -- — '— t< its lowest terms* Jns, — 

5. Reduce -r ; -— - to its lowest terms 

c3 — a*x-^ ux- f-^-3 



0. Reduce -- — —r-- rrT— n — : to us lowest i 

a^x + '2a^x^-^^:ux^+x^ 



36 FRACTIONS. 



PROBLEM VI. 

To add fractional quanttUea together. 



RULE. 



1. Reduce the fractions to a common denominator, a 
in problem the third. 

3. Add all the numerators together, and under thei 
sum write their common denominator, and it will give th 
sum of the fractions required. 



EXAMPIES* 

\, Having — and — given, to find their sum. 

jrX3=3x 
xX2=:2jr 

3X3=6 

3jc 2x 5x 

1 = — =s Bum required^ 

6 6 6 

a c € 

2, Having — , — , and -— given, to find their sum. 
^ b d / ° 



axdyf:ii»adj 
€Xdxf=^cdJ 
e xbxdsssebd 



■l-MBMitaMi«M 



bxdX/^bdf 
hdf^ bdf^ b(if bdf ^ ^ 



FRACTIONS. 37 

•3. Let a — — r-" and d+^ — be added together. 
c 

« X «t > ^numerators. 

And bxcsssdc = common denominator, 
^ 3x2 ■ aajc 3rj:3 , ^ , Sa^jtr 

Therefore a }-^H =« 1 ^<^^ — T— = 

^ be be be 

, , . 2abx — 3rx3 . . ^ 

tf+oH — — e=2 sum required* 

be , 

, .,,3ar ,a7 , . I5x+2d.t' 

4. Add —7 and — together. Jlna. -^^-r — . 

2b 5 lOd 

X X X .'V 

-5. Add — • — 1 and — together. jina. x+—., 

2 3 4 1 -i 

6. Add -- — and — • together. Ans — . 

3 7 21 

7. Add j:H to 3x-\ ^na. 4x+^ . 

3 4 12 

tijr2 X \ a 

^« It is required to add 4x, -— -, and-- — together, 

t&d ix 

fix Tx 2.z*-4- 1 

^' It is required to add --, ^7, and — -7 — togetiur. 

3 4 D 

*0. It is required to add 4jr, -^ , and 2+ -7- lege the ly 

^^ It is required to add Sx-] and x ~ together. 



^^ In the addition of mixed quantities, it is best to bring the frac 
?|^ parts only to a common denominator, and to affix their sw^ to 
w tvun of the hitegers, interposing the proper sign. 

D 



ZS FRACTIONS. 

PROBLEM VII. 

To BublrcLct one fractional quantity Jrom ai 

RULE*. 

1. Reduce the fractions to a common dei 
as in addition* 

2. Subtract the numerators from each 
under their difference write the common dei 
and it will give the difference of the fractions r 

EXAMPLES. 



I, To find the diffcrenceof — - and --:. 

3 11 

xX11«11j7 
2xX 3= 6x 



3xlls33 

Wa: 6a: 5x ,._ 
— — — 5=»— =s differ enct 
33 33 33 -" 

3.^ To find the difference of *-----* and 



3d 5c 

(x— a)X5caB5cx — Sac 
(2fl— 4x) X 3A SBB 6fl* — 1 2bx 



ZbX5c^\Sbc 
Scx-'-^ac 6<7A— 12*x 9jrx— -Sao— 6ad4 

\bb~ "^ Tibe • i5cb 

i^erence required. 



* ne tame rule maj be observed for mixed ^uan 



FRACTIONS, 39 

^\2x .3x . 39:c 

3. Required the difference of--p and y. ^ns. — . 

4. Required the difference of 5y and — . ^ns. — . 

5. Required the difference of— and — . ^/i«. -^. 
d. Required the difference between — ^ and — . 



tdnSt 



bd 



Zx-^a - 2x+7 
T, Required the difference of — — - and 



Ann* 



5d 8 

24jc+8fl — \Obx — 35^ 

406 



X • X^'^O 

?. Required the difference of 3a: + -— " and a;— . 

. , cx+bx+a5 

Ana.^x-i r- 

be 



PROBLEM VIII. 

To muUifify fractional quantities together » 

RULE*. 

Multiply the numerators together for a new numera- 
^) and the denominators for a new denominator, and it 
^ give the product required. 

* t When the numerator of one fraction, and the denominator of 
^ other, can be idivided by some quantity which is common to each> 
^ <IQOdents may be used instead of them. 



40 FRACTIONS. 



EXAMPLES. 

1. Let it be required to find the product ol 

2. Required the product of—, — , and . 

■ * 2 5 21 

a: X 4a: X 1 Oa: > 40 jt^ 4x^ 

S. Required the product of— and . 

^ a a + c 

Sjc Sa 

4. Required the product of — and ---. ^ 

5. RequH'ed the product of — and . j1 

6. Find the continued product of — , — j and 

a c 

A 

7. It is required to find the product ef />-{ a 

a 

Ans, 



2. When a fraction is to be multiplied by an integer, thi 
is found by multiplying the numerator by it ; and if the ii 
the f»ame with the denominator, the numerator may be take 
j)roduct. 

3. When a fraction is to Jbe multiplied by any quantity, 
same thing whether the numerator $e multiplied by it, or I 

minaior divided by it. ' 



FRACTIONS. 41 

d. Required the product of — - — and ■ « 

9. Required the product of xH , and — tj« 



PROBLEM IX. 

To divide one/ractional quantity by another, 

auLB*. 

Multiply the denominator of the divisor by the nume- 
rator of the dividend, for a new - numerator ; and the 
Qumerator of the divisor by the denominator of the di^- 
videndy for a new denominator. 

Or, which is the same thin^, invert the divisor, and: 
proceed exactly as in multiplication* 

EXAMPLES* 

♦« Required the quotient of -"-divided by — . 

X 9 9x 3 

-r-X'T-asr— =T= \\ ^a^ quotient required,^ 

3. Required the quotient of — divided by —^ 

Sa d 2ad ad . ^ 

— X — « —7- = :r7— = quotient required. 



* 1. If the fractions to be divided have a common denominator; 
^^ the numerator of the dividend fior a new numerator, and tte^ 
"^merator oi the dxviior fbr the denominator. 

D 3. 



j. 
k 



. I 



I 



42 FRACTIONS. 

3. Find the quotient of 7- divided by *^ 



2x — -Zb ' Sx+a 

x+a 5x+a Sx^+^ax+a"^ . ^ 

^ .. . /. = — o..« OA - . = 5'"orz>«f required. 



V.r — 26 «t'-fZ> 2a-2 — 2^2 



2r2 X 

J - 4. Find the quotient of -rr- — divided by 



; 2x2 ^x+a ^x^x^x+a) 2x 

— 7~ — - X = > — : — -= =:nuone: 

5. Let — be divided by — . -^^«. -r: 

fi. Let --- be divided bv 5x. jim, - 

7 . • 3 

7. Let -r-— be divided by — • ^««. / — 

i 6 ' 3 4x 

J? j^ 2 

i 8. Let be divided by — . jins. 



X' 



I 

y. Let — be divided bv — . jins. ^ 



y. Let — be divided by —r, jins. ^ 

3 ob %a 

jc*~""6 3cjr JT"*^ 

>0. Let be divided by —7 . Ans, — -— 

8crf ^ 4rf 6c*x 



I x^— ^4 x2+dx 

U. Let — »; • 7- be divided by ^• 

I x^ — 2bx-^b^ ' X — 



2. Wheil a fraction is to be divided by any (quantity, it is the 
thing whether the numerator be divided by it, or the denominito 
multiplied by it. 

3. When the two numerators, or the two denominators, canb 
divided by some common quantity, that quantity may be thi«|| 
out of each, and the quotients used instead of the fractions £ 

proposedL * 






[ 43 J 



INVOLUTION. 

Involution is the raising of powei*s fram any proposed 
i^t; or the method of finding the square, cube, biqua- 
^fate, 8cc. of any given quantity. 

RULE*. 

Multiply the quantity into itself as many times as 
.^here are units in the index less one, and the last pro- 
'^Mct will be the power required. Ovy 

Multiply the index of the quantity by the index of the 
l^ver, and the result will be the same as before. 

^ote. When the sfgn of the root is +, all the powers^ 
|ofitwili be +; and when the sign is — , all the even. 

)wcr!i will be +, and all the odd powers — . 



0)1 



tJ 



EXAMPLES. 






5;-;; I 



a*=a9guare 
a^ssicube 
'a*ssi4ih power 
a'=zSth.povifii9 



a^ssaguare 

a^j root > n .^. ^ 
' ra^s=s4in power 

a^^ssSth power 



I, 



^ + 9a^=aguare 

-^a,root^^ Sia4^^tk power 
J — 2^3a'^Bs»Sih power. 



sr 



u * The nth power of any pradnct is equal to the nth power of 
JSch of the factors, multiplieci together. 

(^And the nth power of a fraction is equal to the nth power of the 
^H^eratori divided by the nth power of the denominator. 



44 



1 - 









fi.T^Ot 




^ as biquadrate 






40*** -s- agtiarc . 






tfOOt 






.3 



^+a«*«5««'' 






«r 






sscuB 



f 



INVOLUTION. 45 

EXAMPLES FOR PRACTICE. 

1 . Required the cube or third power of 2a^> 

Ana, 8a«. 

2. Required the 4th power of 2a2 jr. Ane, l^^xK 

3. Required the third power of — %x^y^, 

Ana. — 5l2x6i/0* 

~, - 2a^x \6cfix^ 

4. To find the biquadrate of -rr» Ana. ^ . , o > 

^ 3^2 8168 

5. Required the 5th power of c — x. . . 

Ana, a* — Sa^x+lOa^x^ — lOa^x'-fSax^ — x^. 

Sia ISAAC NEWTON^s Rule for raiaing a binomial 
or reaidual quantity to any power whatever*. 

I 

% 

'. To find the terma without the co^effiqienta. The in- 
*^f ^ of the first or leading quantity begins with that of the 
^*^en power, and decreases continually by 1, in every 
^^^^ to the last ; and in the following qikntity the in* 
^'^es of the temis are 0, 1,2, 3, 4, &c. 

^. To find the uncia, or co^efficienta. The first is al- 
.^*ys 1, and the second is the index of the power : and, 
'^ general, if the co-efficient of any term be multipUed 
T }ht index of the leading quantity, and the product be 
l^^ided by the number of terms to that place, it will give» 
^ co-efficient of the term next following. 



This rule, expressed in general terms, is a|^Uows : 



a 



n 



^. 



„«.! n— 1 n— 2, . WPh— 1 #»— 2 n— 3 

5B=a -f-n .a 6 + " • "^ ^ 6* + « . -5— . — r- a 

r^---n n It— 1, . «— 1 n— 2,^ w— 1 n— 3 n^3 

o ssa — n.a b-f-n. -^r- fl b^ — «•"■«- • "rT ** 



46 INVOLUTION. 

Kote, The whole number of terms will be one 
than the index of the given power; and when both 
of the root are +9 all the terms of the power will I 
but if the second term he — « all the odd terms will I 
and the even terms — - 

EXAMPLES. 

1. Let a-\-x be involved to the fifth power> 
The terms without the co^efficienta will be 
fl*, a^Xy Q^x^^ a^x^^ ax^f a:*, 
and the co^tfficitnts will be 

5X4 10X3 10X2 5X1 

^nd therefore the fifth power i% 
«'+5a<x+ iOa3x«+ 10a*j?3+5ax^+x^ 

% Let x^— a be involved to the 6th power. 

The tetm^without the co^efficients will be 

a?*j x*o, jHo>, jc^c*, x*a*, xa^j fl*| 

and the co^efficientB wiil be 

6xS 15X4 20X3 15X2 6X1 
h 6, -J-, -j~, -J-, -J-, -g-, 

or I, 6, 15,20, 15, 6, 1. 
t^nd therefore the 6M fiower ^o:— ^ « 

j?a«^jp5a^-i5j;4fla — 20x3a3+l5x2a4— 6ara; 

^ Required the 4th power of x— ^. 

.^n*. x^— 4x3fl+6j?*a**-4j:a' 
4. Required the 7th power of x+c. 
./^««. j77+7jr6fl+2lj?«a3+35x4a3+35x3a* + 21x 
7jra*4-a7, ^ 

' ' • ^ . ■ ,LirV ■» 

The sam of the coefficients, in every power, is equal to th< 
bar 2, imised to that power Thus, l+laB2 for the first i 
14-2+l«4«2» for jAe.squaiej l++33+la:8«s2J f 
euM^ or thicd power^||B/ - • 



J 



J 



£ 47 ] 



EVOEUTION. 

Evolution is the reverse of involution, or the method 
^ finding the square root, cube root, Sec* of any given 
^IKiantityy whether cimple or compound. 

CASE I. 

To find the roots of simple quantities. 

RULE*. 

Extract the root of the co-efficient for the numerical 
put, and divide the index of the letter^ or letters, by the 
index of the power, and it will give the root required. 

EXAMPLES. 

1* Required the square root of 9^S And the cube root 
of 8x3. 

^no* v^9^'asSjr|aB3x; and •^8:p^83x|sa3x. 

. ^ 3x*y* 

3. Required the square root of ' » and the cube root 

27a^ 

3r«f^ xy ^^ , A6x^^ 2xy ^. - 



* Any even root of m affinnative quantity may^ be ekher + 
«.— {- thus the square root of ^ «t it either ^ a, or i— a : for 
(+«)X(+fl)— +a». «nd (-.fl)x(— «)— +A 

And an odd- root of any quantity will have the same eign as 
fhc q^ianidty ittk^: thw the onbe root of ..\ufi it ^randtho eube 



y^ 



48 EVOLUTION. 

3. Required. the square root of 3a3x<^. Ann^ax^y/ 

4. Required the cube root of — 125a3x^» An9. — 5a j 

5. Required the square root of — j- . An9» ^ — = 

6. Required the 4th root of 3560^x8. Ans. 4a:i 

7. It is required to find the 5th root of — SSx^yi®. 

Ana. —2x1 



CASE II. 
Tojind the square root of a compound qttantity, 

RULE. 

1. Range the quantities according to the diraenaio 
of some letter, and set the root of the first term in tl 
quotient. 

2. SubtracUthe square of the root, thus found, fro 
the first term, and bring down the two next terms to tl 
remainder for a dividend. 

3. Divide the dividend by double the root, and set H 
result both in the quotient and divisor. 

4. Multiply the divisor, thus increased, by the ter 
last put in the quotient, and subtract the product fro 
the dividend ; and so on, as in common arithmetic. 



Trootof — a3 is -^; for (+a)X(+fl)X(+fl)=«+fl^and(-< 

Any even root of a negative quantity is impossible : for neftH 
{4-a)X(+«) nor (— «)x(— «) can produce — . 

The nth root of a product is equal to the nth root of each ff fjj 
factors multiplied together. « 

And the nth root of a fraction is equal to the nth root of th»f| 
joentor, (fa'vided Iby the nth lOot of the denominator. ' . ^ 



EVOLUTION. 40 



EXAMPLES. 



1. JBxtract the square root of x^— 4jc3+6jJ'aL--4:c+I. 
074 — 4^:3+6^:2 — 4:c+l(x2— 2a:+l=: root. 



— 4jr3-f^4j7* 



2x^ — 4x+ 1 



2. Extract the square root of 4a4+ \2ii^x.+ ISa^ir* 

4fl4-{- i2a3jK.-f I3a2j.2-|-6ax3+x4(2a«+3(rx+x2 
4fl4 



4c^+ 3aar) 1 2c«a: + 1 3a>:r3 
42a3x+ 9a2x8 



4o2+6ax+x2) 4ta^x^+6ax^+x^ 



3. Rcquu'ed the square root of fl'* + 4a3x + 6a?a:* 
+Aax^+x^. Jina. a^+^ax,+x^. 

4. Required the square root of x^o-^x^+^x^"^ 
X ^ 

2 ^16 ^ 

5. It is required to find the square root of a«+ x^. 

2a 8a3 vl6a* 
E 



50 EVOLUTION. 

CASE III. 

Tojind the roots o/fiowera in general, 

RULE*. 

1. Find the root of the first term, and place it 
quotient. 

2. Subtract its power from that term, and bring 
the second term for a dividend. 

3. Involve the root, last found, to the next ] 
/power, and multiply it by the index of the given 

for a divisor. 

4. Divide the dividend by the divisor, and the 
tient will be the next term of the root. 

5. Involve the whole root, and subtract and 
as before ; and so on till the whole is finished. 

EXAMPLES. 

1. Required the square root pfc^ — 20^^17 + 



2a«)— 2o3jr 
a*— .SflSjc^ c2x2 



2a«)2aax« 



a^ — 2a^x+Za^x^ — 2ax^ + x^ 



As this ni£thod, m high powers, is generally thought tc 
!!^ll "*^ "^' ^ improper to observe, that the roots of cor 



I 



EVOLUTION. 5 1 

2. Extract the cube root of x»+6x«— 40x3+96x 
— 64. 

a:« + 6x« — iOx^+96x — 64(a;2+2x— .4 



3x4) 6x* 



x«+ 6x5 +12x4+8x3 



3a74)««i2x4 



x«+ 6x«— 40073^+ 96 jp— 64 



3. Required the square root of ««+2a3+2flc+42 
+2bc+cK jina^a+b+c. 

4. Required the cube root of x^ — 6x«+15x<— -20 
ar5+}5x»— 6x+l. .^n«. xa— 2x+l. 

5. Required the biquadrate root of 16a4 — 96a^x 
+ 2 1 6a2x2— 2 1 6ax3 + 8 1 x^. Jn8. 2a— .3x. 

6. Required the" fifth root of 32x« — 80x4+80x3— k, 
40a:2+iOx — 1. Jns. 2x— 1. 



1- Extract the roots of some of the most simple tenns, and con- 
'Jttit them together by the sign + or — , as may be judged most 
^iritable for the purpose. 

3 Involve the compound root, thus found, to the proper power, 
^^^» if it be the same with the given quantity, it is the root re- 
quired. 

3. But if it be found to differ only in some of the signs, change 
^^m from + to — , or from — to +, till its power agrees with 
the given one throughout. 

Thus, in the fifth example, the root 2a — 3x, is the difference 
(jf the roots of the first and last terms ; and in the 3d example, the 
root a-^-b-{'C is the sum of the roots of the Ist, 4th, and 6th terms. 
The same may also be observed of the 6th example, where the root 
i» found from the first and last terms. 



[ 52 ] 

SURDS- 

Surds are such quantities as have no exact root, be 
usually expressed by fractional indices, or by mean 
the radical sign \/ placed before them. 

Thus, 2', or \/2, denotes the'square root of 2, anc 
the cube root of the square of ^ ; where the numen 
shows the power to which the quantity is to be rais 
and the denominator its root. 

PROBLEM I. 

To reduce a rational quantity to the form of a surd 

RULE. 

Raise the quantity to a power equivalent to t 
denoted by the index of the surd, and over this i 
quantity plaee the radical sign, and it will be of 
form required. 

EXAMPLES. 

1. It is required to reduce 3 i% the form of 
square root. 

\,Firat^ 3x3sbs3*=9; vyhenccs/^ the ansz 

2. It is required to reduce 2x« to the form of 
cube root. 

I^irttt^ 2arX2j7»x2a:2=:(2x'>)3=8xS; whence ^8x 

(Sx*) !» the answer » 

3. Reduce 5 to the form of the cube root. 

Jna, (125 

4. Reduce hxy to the form of the square root. 



SURDS. 53 

1 

5. Reduce 2 to the form of the 5th root« ^na, (32)^. 

1 

6. Let d^ be reduced to the form of the 6th root. 

7. Reduce a + b io the form of the square root, and 
a — 6 to the form of the cube root. 

PROBLEM ir. 

To reduce quantities of different indices to other equiva- 
lent ones that s/iall have a common index. 

RULE. 

1. Divide the indices of the quantities by the given 
index, and the quotients will be the new indices for those 
quantities- 

2. Over the said quantities, with their new indices, 
place the given index, and they vill make the equivalent' 
quantities required. 

3. A comtnon index may also be found by reducing 
the indices of the quantities to a common denominator, 
and involving each of them to the power denoted by its 
Dumerator. 

EXAMPLES. 

1. Reduce 15* and 9^^ to equivalent quantities having 
the common index i. 

1 1 1 2 2 1 . , . 

— -» s — X — = — « — «Ui index* 

4 2 4 1 4 a 

1 1 1 2 2 1 «... 
L.^Bs X — sss— OB — n2(/ index*. 

6 2 6 1 6 3 

Thertfore{\5^)^ and (9^)*— quantiHcM re^uire<t:^ 

R 2 



54 SX^RDS. 

2. Reduce o^ and x-^ to the same common index ^ 

"2*3 1^1 I 

5 = — X — ==— =2a index, 

4 3 4 1 4 

1 1 

Therefore {aP'^ and (^-fp = quantities required, 

* 3, Reduce 3 and 2"^ to the common index -J. 

I 
Ans, (27)'^ and ( 

4. Reduce a* and 6^ to the common index |. 

1 

Ans, {'i^j^ and (.' 

i , 1 
5* Reduce a** and A"* to the same radical sign. 

Ans. "^x^a"* a?id '""v 

i ^ 

6.. Let (<! + *) and (a— *)J be reduced to a comi 

index. 

7. Let (a + ^)^ and (a*— d)* be reduced to a comi 
index. 

PROBLEM IIL 

To reduce aurda to their moat aim/ile terms. 

RUI/E*. 

Find the greatest power contained in the given si 
and set its root before the remaining quantities, uith 
proper radical sign between them. 



* When the given surd contains no exact poweri it is alread 
MtM mam tkB^mmx 



SURDS. 55 



EXAMPLES. 



1. It is required to reduce ^/48 to its most simple 
terms. 

v/48=v/(l6x3) =?v/ 16 Xv/3=4X x/Srsfv^^, the 
answer. 

'2. It is required to reduce -^108 to its most simple 
terms. 

^VIOS =:^ (27X4) =^27 X^K4 = 3x ^4 =3 ^4, 
■'/fcf answer. 

3. Reduce \/125 to its most simple Jt^rms. 

50 

4. Reduce \/ t—— to its most simple terms. 

147 

5. Reduce -^243 to its most simple terms. 

^/i*. 3^9. 

16 

6. Reduce -^ — to its most simple terms. 

81 * 

Jns. 1^18, 

7. Reduce x/9Sa^x to its most simple terms. 

8. Reduce ^(x^— a»jp») to its most siipple terms. 

9. Reduce (fl3jc+3a3jr«>J to its most simple terilRS. 

10. Reduce (32afi — 96a^a;)J to its most simple terms. 



3* 

56 SURDS. 



PROBLEM IV. 

To add surd quantities together. 



^ 



« 



^ 



RULE. 



1. Reduce such quantities as have unlike indio 
ether equivalent ones, having a common index. 

2. Bring all fractions to a common denominatofi 
reduce the quantities to their simplest terms, as ii 
last problem. 

3. Then, if the surd part be the same in then 
afiuex it to the sum of the rational parts, M'ith 
sijg;n of multiplication, and it ^ill give the total 
reqAiired. 

But if the surd part be not the same in all the qt 
ties, they can only be added by the signs + and — . 

EXAMPLES. 

1. It is requh*ed to add y/27 and ^48 together. 

Firsts -v/27=v^(9X3)=3-v/3; owcf ^^48 =v/( 
a)=r:4v'3; 

Whence, 3v^3+4v/3=(3+4)^3»7v^3=i aui 
quired. 

2. It is required to add -^500 and ^108 togethe 

Firat, v^ 500 =^ ( 125 X-4) ssS^4, and^ 108 » 
(27X4)«3^4; 

fVhencej 5^4+3^4w(5+3)^4»8^4«, ««« 
quired. 



SURIJp. . 57 

3. Required the sum of ^72 aiid ^/128. 

Ana. llv^2. 

4, Required the sux^^jf v^ 2 7 and ^147. 

Ana. lOV'S. 

' An8,\iy/6. 



^^j^biA- «i^/» 



6. Reauirw^ tl^ sum of ^40 and -^135, 

7. Required the sum of 3^1 and S^V^V* 

8. Required the sum of l^a^b and %^^Xbx^. 

9. Required the sum of 9^/243 and 10v^363. 

10. It is required to find the sum of a** and a"» . 

1 1. Required the sum of ^Ita^x and y/Za^x, 



PROBLEM V, 

To auhtract^ or find the difference of^ surd quantiUes. 

RULB. 

' Pi'epare the quantities as in the last rule, and the diffe- 
rence of the rational parts annexed to the common sard 
will give the difference of the surds required. 

Hut, if the quantities have no common surd, they can 
onlr be subtracted by means of the bign — s 






58 * Sd^RDS. 



EXA.MPLES. 



1. It is required to find the difference of v^.448 and 
Firsts V'448 = %/(64 X 7) Si^r ; and v' 1 12 = v/ 

Whence 8v>^l^4x/7. = (8« — •*)v'r = 4vr'7, atjperencc ^ 
required. 



1 



2. It is required to find the diffwrcnoc of lor^^ nnd 

24* ''^^ 

Firsty 192^= (64X3)*^ =4.3*; flwtf 24* « (8X3)^=s " ' 
1 
2.3^. 

11 11 

Whence 4.3^— 2.3^ = (4 — 2)^.3 == 2.3'3^=x: difference 

required* 

3. Required the difference of 2v^50 and v^l8. 

An9, 7-v/2. 

1 1 

4. Required the difference of 320^ and 40"^, 

Ana, 2 ^5 . 

3 5 

5. Required the difference of \/— and %/--. 

6. Required the difference of ^-f and ^-5^. 

7. Find the difference of y/^Oa^x and ^/20o2x3. 

^/i*. (4a2— 2(w?) \/5:c. 

8. Required the difference of S^a^A and -^a^b. 

1 1 

9. It is required to find the difference of x^ and x"* . 



SURDS. 59 



PRjgBLEM VI. 



To multijily surd quantities togtAler. 

^f^^ RULE. 

Reduce the surds to the same index, and^fteproduct 
efthe rational quantities annexed to the pit>duct of the 
^ surds will give the whole product required ; which may 
be reduced to its most simple terms by Problem 3. 

EXAMPLES. 

1. It is required to find the product of 3 V' 8 and, 

i Here, 3 X2 X v'SX v'S =6^/(8X6) = 6 v^48:=6 v" 
(16X3)=6x4X\/3=24v/3= /irorfwc/ required, 

2. It is required to find the product of ^^f and 
13 2 5 3 .10 3 ^15 5 I 

3 1 

X^IS = — ^ 1 5 = - ^ 1 5 = firoduct required. 

3. Required the product of 5v/8 and 3-^/5. 

jins, 30^/ It). 

1 2 

4. Required the product of — \/6 and— ig^lS. 

./fw«. -^4. 

2 1 3 7 
i^. Required the product of -—v/—- and -7-^/— . 



60 SURDS. 

/ 6, Reqviired the product of -^18 and 5-^4. 

7. It is reqjiiced to find the product of <z* and a "3^. 

- • An8*{a^yor 

1 1 

8. Required the product of (jt+j/)^ and (^x+y)"^. 

9. Re<K|iABd the product of x+^y and x — ^y« 

10. Required the product of (a+\/^)^> and (a- 

JL 1. 

11. It is required to find the product of x« and x^. 



PROBLEM VII. 

To divide one surd quantity by another, 

RULE. 

Reduce the surds to the same index, and the quotie 
of the rational quantities being annexed to the quotie 
of the surds, will give the whole quotient requirec 
which may be reduced to its most simple terms 
before. 

EXAMPLES. 

1. It is required to divide 8^108 by 2^6, 
(8-5-2) v'(108-^6)=4^18=4v^(9x2) =4x3^/2= 

12^/295 quotient required. 

2. It is required to divide 8^512 by 4^2. 

Ill 1 > 

8-i-4=a:2, and 512' H-2'^=:256"^sB=4.4'^ ; 

Therefore 2x 4x4'y« 8.43= quotient required. 

^. Let €%/lO be divided by 3>/5, An9.2y/i 



SURDS. 61 

4. Let 4^1000 be divided by 2^4. Ana, 10^2. 

3 1 2 1 1 

5. Let— v/—- be divided bx — \/— . Ana, -r-v^3. 

4 135 ,35 o 

K *i 2 3 25 

6. Let --^— be divided by — ,3/—. Ana ^3. 

7 3 '54 21 

2 2 1 31 8 > 

7. Let — \/a. or — a^. be divided by — a"^. •^w*. — a"*^. 

5 5 ' 4i '%. 15 

8. Let the quantity x^ be divided by the quantity x^ . 

9. Let x* — ^jcrf — b+di/b be divided by x — ^/b. 



PROBLEM VIIL 

To involve aurd quantitiea to any fiotoer. 

RULE. 

Multiply the index of the quantity by the index of the 
power to which it is to be raised, and to the result annex 
the power of the rational parts, and it will give the power 
required. 

EXAMPLES. 

2 1 
1. It is required to find the square of -—a''^. 

Firat, r±)2^±x — = — , and Ca'^y=za'^ '^ =a^ 
3 3 3 9 ^ 

2 1 4*4 

Whence (— a^)* =*"^^*)^ =* — ^a* = aguare requirei. 
o y «f 



/ 



62 SURDS. 

5 

It is required to find the cube of — v'''* 

FiTBU (1)3= Ix-i xf =^, and (7^)3 =7^ =(73)5; 

5 12^ ^ 125 ^ 

M^A^wcf (— v^7)3 = — • (73)^ = — • (343)^ ^cube re^ 

quired, 

. 3. Required the square of 3^3. ^n*. 9-^9. 

1 

4. Required the cube of 2^ or -v/2. ^n«. 2^2. 

1 * I 

5. Required the 4th power of — v'6. -— . 

6.x 36 

6. It is required to find the wth power of a*" . 

7. It is required to find the square of 3 + V'S. 

8. It is required to find the cube of 2x — 3^y. 



PROBLEM IX. 

To extract the roots of surd quantities^ 

' RULE*. 

Divide the index of the given quantity by the index 
of the root to be extracted, and to the result annex the 
root of the rational part, and it will give the root re- 
quired. 



• The square root of a binomial or residual surd a + b, or 
A — B, may be found thus : take \/( a^ — b^) = d ; then \/{a. 

Thus the square root of 8+2\/7=l+ \/7, and the square root of 

2 — \/8=\/2— 1; but for th? cube, or any higher root, no general 
rule can be given. 



SURDS. . 63 



EXAMPLES. 

1. It is required to find the square root of 9^3. 

1 1 

WhencCyi^^Z^isi^.^ ^s^square root required, 

2. It is required to find the cube root of — v^2. 

First, ^-5-=-, aTzrf (^2)^=2^ ' =2^; 

1 I- 1 1 ' 

Whence^ ( — ^2) = — 2^ =:cube root required. 

3. Required the square root of \0\ Ana. 10^/10. 

8 2 

4. Required the cube root of ;—a3. Ana.^, 

1 1 

5. Required the 4th root of 3jc^. ./fn«. 3^. x^. 



X 



^ 



6. It is required to find the 72th root of x»». 

7. Required the square root of j?* — 4iXx/a+4ia. 



INFINITE SERIES. 

jtn irifinite aeries is formed from a vulgar fraction, 
having a compound denominator, or by extracting the 
root of a surd quantity ; and is such, as, being con- 
tinued, would- run on ad infinitum, in the manner of a 
decimal fraction. 

But, by obtaining a few of the first terms, the law of 
the progression will be manifest, so that the series may 
be continued) without actually performing the whole 
operation. 



64 INFINltE SERIES. 

PROBLEM I. 

To reduce fractional quantUlea into injinite series*. 

Divide the numerator by the denominator, as in 
common division; and the operation continued, as far 
as may be thought necessary, will give the series re- 
quired. 

EXAMPLES. 

ax . . . . . 



a — X 


uc 


piupuac 


;u Lvi 


UC l«UIIY< 


infinite series. 




, 






a-— a:)ax...( 


x-\- 


a a^ 


xA 
' a3' 


&c. 


aor— 


-X» 


9 








X^ 




/ 


X^- 


^3 






■ 


a 










x^ 
a 










x^ 


x^ 






a 


■u» 








07* 




. 






a» 










07* 


X* 


- 


♦ 




a*"" 


■^o^ 






x^ 



rNFINITE SERIES. 65 

2. Let — : — be converted into an infinite series. 

1+X)1 (1 X + X^ X^y &c. 

l-i-x 



'X — x^ 



x^ 



x^-{-x^ 



'X^ . 

-x^~^x^ 



x^ 

I 

. Let be converted into an infinite series^ 

a-^x 

b ^ x^ x^ 

An8.—X^\ +— r+, &€.>; 

a a a^ a^ 

4. Let be converted into an infinite series. 

a — X 






hi X x^ \ 

Jins. — x( 1+— +-^» &C.1 

a \ a a^ / 



5. Let be converted into an infinite senes. 

1 — X 

a* 

6. Let r -^ be converted into an infinite series* 

jina. 1 +-r 7»^^- 

a a* «t* 

r. Let -, or its equal be converted into an inv- 

2 ^ 1+1 

finite series*. 

E2. 



6 INFINITE SERIES. 

PROBLEM 11. 

To reduce a compound surd into an infinite series. 

.auLB*. 

Extract the root as in common arithmetic, and the 
operation 9 continued as far as may be thought necessary 
will g;ive the series required. 

EXAMPLES. 

I. Extract the square root of a^+x^ in an infinite 
series. 

^, . a:« x^ , x^ Sx^ ,_^ 

^ ^ ^2a 8o3^ I6a« 1280^' 

2a+— )x8 
2a' 

2 1 ^^ 
^ 4a2 



x^ x^ x^ 



X* X* X"' 



0«il*t 



4a2 Sa-* • 64a» 

x* x^ ^ ^ x^ x^ 

2a+ — — TT) ^^0 



4a3' ' ga-* 64a<» 
x^ x^ 



8a* 16a» 



5x8 



64fl«' 



* This rule is chiefly of o^e in extracting the square r 
apention Iwing too tedious for the hi£;her powers. 



^ INFINITE SERIES. 67 

2. Let ^^ 1 + 1 be converted into an infinite series. 

1111 

'2 8 ^ 16 32 



3. Let \/fl2— .x* be converted into an infinite series. 

07* X* x^ ,_, 

2a 8a3 i6a« 



4. Let -^1 — wc3 be converted into an infinite series. 

. ^ x^ ^fi^ Sx^ ,_ 

3 9 8l' 

5. Let ^/a^+b be converted into an infinite series. 



PROBLEM III. 

^0 reduce a binomial surd into an injinite series ; or t9. 
extract any root of a binomiaU 

RULE*. 

Substitute the particular letters of the binemial with 
f^eir proper signs, in the following general form, and 
\ ^JH give the root required; observing that p is 

^"G first term, q the second term divided by the first, 

m 

^ the index of the power or root \ and a, b, g, d, &c. the 

*o^cgoing terms with their proper signs: 

' -f qp« = P„ (a) + -jj- Aq(B) +-^ Bq(c) + — 3^ 

Cq(D) + !:iZl5:?Bq(B),&c. 



Any surd may be taken from the denominator of a fraction and 
'^ in thi 
index. 



f}^^ in the numerator, and vice versa, by only changing the sign of. 



68 INFINITE SEKIES. 



EXAMPLES. 



1. To extract the square root of r^ — x^, in an infinite 
series. 

/ x^ m 1 

Here P=srS qs=:_— -, and — = — ; therefore (r* — 

-'7=r+ (± X A X-f!) + (-^ X » X -5) + (- 
_XCx--j + (- JXDX--J, t^c. = r+ (__ 

A H rB H -c +'r-rD> ^c. which, by restoring^ the 

4r^ 6r* or* 

x^ x^ x^ 
values o/aj b, c, d, ^c. becomes ^— ^j; r-^- — -rT^ — 

5078 



l28r^ 



, Isfc, = series required. 



2. To find the value of ; — ; — -, or its equal (a + b)-^^ 

(a -f- d)^ 

in an infinite series. 

b m —2 
Here p=b a, qs=— , and — = ; therefore (a + by*' 

--+(-f'4)+(44)+K4)+(- 

f 5b \ .. \ 2b 3b 4b 5b ^^ 

^\ 4a r a^ a 2« 3a 4fl * 

which, by restoring the values of a, b, c, d, becomeM: 
1 23 3d 4b sb . ^ . ^ 

a2 o3 o* tt« ^ao • ^ 



INFINITE SERIES. 69 

3. To find the value of — : — , in an infinite series. 

jins. r — xA 1 , ^c. 

i. To find the value of-- — '-^i in an infinite series. 

I x2 3x4 \5x^ _, 

5. To find the value of;: — ; — -- in an infinite series. 

2x 3x2 4j;3 5j;4 

6. To find the value of (a*+*)^ in an infinite series. 

b b^ b^ 564 

^2a 8a3 lea* 128a7' 

7. Find the value of (a^ — x2)T in an infinite series. 

•*»•• "^ X ' 1 - t:;!- 21;?- i2ir»' ^'- 

8. To find the value of (a^ — ^)^in an infinite series. 

b b^ 5/>3 1064 

jins.a — —^ — ——^^^ — .^^j5^ , l:fc . 

9. Required the square root of — JL — m an infinite 
series. " •* 

X* X4 X® 

a^ 2a4 "^ 2a0 

10. Required the cube root of — f in an in- 
finite series. (as+x^)^ 

1 . 2x* 5x4 40x« ,^ 

Ana. — ;X : 1 1 ^> ^f- 

of 3a2 ^9tt4 81a6' 



70 ARITHMETICAL PROPORTION. 

1 1. Required the value of ^ in an infinite 

series. " ax-^x 

^ X :r2 ^4 ^5 

^ne, j iSfc* 

a a^ a* a* 



ARITHMEICAL PROPORTION. 

Arithmetical firofiortion is the relation which two quan- 
tities, of the same kind, bear to each other, with respect 
to their difference. 

Four quantities are said to be in arithmetical firofiortioriy 
when the difference between the first and second is equal 
to the difference between the third and fourth. . 

Thusy 3, 7, 12, 16, and a, c+A, c, c+6, are arith* 
metically firofiortionaU 

Arithmetical firogreaaion is when a series of quantities 
either increase or decrease by the same common differ- 
ence* 

Thus^ 1, 3, 5, 7, 9, 11, b'c. and a, a-f-^i a'\-2by 
fl+3^, a+4d,^a+5A, iSJ'c, are series in arithmetical fir o» 
gression^ whose common differences are 2 and b. 

The most useful part of arithmetical proportion is 
contained in the following theorems : 

I. If four quantities be in arithrhetical. proportion, the 
sum of the two means will be equal to the sum of the 
two extremes. 

Thusy if 2^ 5, 7, 10, a?id a, d, r, d^ are in arithmC' 
tical ftrofiortiony then will 2 + 10=5+7, and a+e/ = 
b+c. 

II. In any continued arithmetical progression, the 
sum of the two extremes, and that of any two terms 
which are equally distant firom them, are equal to each 
other. 



ARITHMETICAL PROPORTION. 7 1 

Thu8^ in the aeries 2, 4, 6, 8, 10, 12, e^c. 2 + 12 = 4 + 
10=6+8. 

III. The last term of any arithmetical series is equal 
to the sum or difference of the first term, and the product 
of the common difference by the number of terms less 

* ODe ; according as the series is increasing or decreasing. 
Thua^ the 20th term of 2, 4, 6, 8, 10, 12, ^Tc. m =2 + 
2(20— .l)n=2+2x 19=2 + 38=40. 

And the nth term ofa^a'-^Xy a — 2a:, a — 3x, a — 4^, 
^cia ssa — (n — l)Xx=a — (n — l)x. 

IV. The sum of any series of quantities in arithme- 
tical progression is equal to the sum of the two extremes 
multiplied by half the number of terms. 

Thuay the sum o/* 1, 2, 3, 4, 5, 6, ^c. continued to the 

(1+20)X20 21X20 ^, ,^ ^,^ 
20th term^ia =i — = — r — = 21x10 = 210. 

And the lum ofn terma of a^ a + x^ a -{- 2x, a + 3x, to 

a-\-mxy ia ss (a+a+mj:)-—=(a+^mar). w = (aH — •— 

x)n. 



EXAMPLES. 

, I. The first term of an increasing arithmetical series 
is 3, the common difference 2, and the number of terms 
20 ; required the sum of the series. 

/^ir«^,3+2x(20—l)=3 + 3x19=3+38=41= aat 
term, 

And(3 + 4\)X — = 44X — = 44X 10 = 440= sum 
^2 2 

Required, 

2. The first teiin of a decreasing arithmetical series 

is 100, the common difference 3, and the nun)* " ms 

34 ; required the sum of the series. 



72 ARITHMETICAL PROPORTION. 

Firaty 100—3.(34 — 1) = 100 — 3.(33)= 100 — 9 

1 = last term. 

34 34 

^«rf(l00+l)x~ = 101X--=101 X 17=17n 

sum required, * 

3. Required the sum of the natural numbers 1, 2 
4, 5, 6, Sec. continued to 1000 terms. 

AnB. 500< 

4. * Required the sum of the odd numbers 1, 3, 5 

9, Sec. continued to 101 terms. Ans* 102 

5. How many strokes da the clocks of Venice, wh 
go on to 24 o'clock, strike in the compass of a day? 

Ana, 3 

6. The first term of a decreasing arithmetical serie 

10, the common difference -|, and the number of tei 
21; required the sum of the series. Ana. 1 

7. One hundred stones being placed on the ground 
a straight line, at the distance of a yard from each otl 
how far will a person travel who shall bring them one 
one to a basket, which is placed one yard from the f 
stone ? Ana. 5 milea and 1 300 yai 



• The sum of any number of terms (n) of the arithmetical se 
of odd numbers 1, 3, 5, 7» 9» &c. is equal to the square (n^) of t 
number. 

That is, if 1, 3, 5, 7, 9, &c. be the n)Knbers, 

Then will 1, 2^, 32, 42, 5*, &c. be the sums of 1, 2, 3, Stc. tern 

For 0+li or the sum of 1 term =1*, or 1 

1+3, or the sum of 2 terms =2*, or 4 

4-f-5, or the sum of 3 terms =3*, or 9 

9-f 7, or the sum of 4 terms =4*, or 16, &c. 

Whence it is plain, that, let n be any number whatcveri the s 
of/i terms will be rfi. 






[ 73 ] 



GEOMETRICAL PROPORTION. 



Geometrical. flroflortion is that relation of two quantities 

of the same kind, which arises from considering what 

part the one is of the other, or how often it is contained 

10 it. # 

When four quantities are compared together, the first 

~ and third are called the antecedentsy and the second and 

i' fourth the conaequents. 

Ratio is the quotient which arises from dividing the 
antecedent by the consequent, or the consequent by the 
antecedent. 

Four guantitiea are said to be flrofiortional when the 
first is the same part or multiple of the second as tl^ 
third is of the fourth. 

Thus 3, 8, 3, 12> and a, ar, *, br are geometrical firo- 
/i^rtionala. . 

Direct firofiortion is when the same relation subsists 
between the first term and the second, as between the 
third and the fourth. 

Thfis 3, 6, 5, 10, and x, axj y, ay are in direct firo^ 
Mrtion. 

Reciprocal or inverse firofiortion is when one quantity 
increases in the same proportion as another diminishes. 

Thuaj 2, 6, 9, 3, and <i, ary br^ b are in inverse firo- 
fiortion. 

A aeriea of quantities are said to be in geometrical 
^Progression when the first has the same ratio to the 
^cond' as the second to the third, the third to the 
fourth, &c. 

Thusy 2, 4, 8, 16, 32, 64, Isfc. and fl, ar, or^, ar^ ai*, 
<ir', t3*c. are aeries in geometrical fir ogreaaion. 



74 GEOMETRICAL PROPORTION. 

The most useful part of geometrical proportion is con- 
tained in the following theorems. 

I. If four quantities be in geometrical proportion, the 
product of the two means will be equal lo that of the two 
extremes. 

Thu8y if^y 4, 6, 12, and a, ar^ by br, be geometrically 
firo/iordonalj then vnU 2x 12=as4X6, and axbrz=bxar. 

II. If four quantities be in geometrical proportion j the 
rectangle of the means, divided by either of the extremes, 
will give the other extreme, 

Thusf ifZy 9, 5, 15, and Xy axy y, ay^ are geometrically 

firofiortionaly then will = 15, and . ss=x. 

3 ay 

III. In any continued geometrical progression, the 
product of the two extremes, and that of any other two 
terms, equally distant from them, will be equal to each 
other. 

Thus, in the aeries 1, 3, 9, 27, 81, 243, i^Tc. 1 X 243 = 
3X81 = 9X27. 

IV. In any continued geometrical series, the last term 
is equal to the first multiplied by such a power of the 
ratio as is denoted by the number of terms less one. 

Thusj in the series 2, 6, 18, 54, 162, ^c. 2x34=162. 

V. The sum of any series in geometrical progression 
is found by multiplying the last term by the ratio, and 
dividing the difference of this product and the first term 
by the ratio less one. 

Thusy the-sum of *^y 4, 8, 16, 32, 64, 128, 256, 512 is 

512x2 — 2 

-—=1024—2=1022. 

2 — 1 

And the sum of n terms of a, ar, ar^^ oH, cH, ^c, to 

ar"~^Xr — a ar^ — a r**— 1 



GEOMETRICAL PROPORTION. 75 

VI. If four quantities, a, 6, c, e/, or 3, 6, 5, 15) be pro- 
portional, then will any of the following forms of those 
quantities be also proportional. 

1 . directly^ a : b i : c : d, or 2 : 6 : i 5 i 15» 
3. inversely^ d : a : : rf : r, or 6 : 2 ; : 1 5 : 5. 

3. alternately^ a : c : : 6 : rf, or 2 : 5 : : 6 : 15. '^■ 

4. com/ioundedly, a : a+b : : c : c+c/, or 2 : 8 : : 5 : 20. 

5. dividedly, a i b—ra ::c : d-^c^ or 2 : 4 ; : 5 : 10. 

6. mixed^ b+a : A— a : : d+ci d-^Cy or 8 : 4 : : 20 : 10. 

7. by multi/ilication, ro : rd : : c : </, or 2.3 : 6.3 : : 5 : 15. 

8. by division^ a-i-r : b-r-r : : c : rf, or 1 : 3 ; : 5 : 15. 

9. The numbers a^ by c^ d are i0ltarmonical /trofiortion, 
when a : diiauibi c Vid, 



EXAMPLES. 

1. The first term of a geometrical series is 1, the ratio 

2, and the number of terms 10; whait^ is the sum of the 

• > . .j . 

series f 

Fir9ty 1 X2«==» 1 X5 12= /a«r term. 

.^512x2—1 1024—1 ,^^^ . . 

jind = = 1023 = aunt required. 

2 — I 1 

2. The first term of a geometric series is ^ the ratio 
■|, and the number of terms 5 ; required the sum of the 
series. 

Fir>.t, 1 X (-1)' =1 X ^ = 4 - last term. 

•^"'' (i—rk^i-)^ ('-- i-)-(T-i56)-^T 

121 3 121 121 . . 

=: X — = = ^ aum required* 

243 2 81X2 162 

3. Required the sum of 1, 3, 9, 27, 81, &c. continued 
to 12 terms. -^^w. 265720. 



76 SIMPLE EQUATIONS. 

4. Required the sum of ^ 7) •}> ^) Tt> ^^' continued 
to 13 tenns. 

5. Required the sum of I, 2, 4, 8, 16, 32, &c. conti- 
nued to 100 terms. 






SIMPLE EQUATIONS. 



An equation is, Vfht% two equal ^antities, di fferently 
expr^pd, are compared togetheTEy means of the sign ass 
plac^Rfetween them. 

Thus 12 — 5=s7 is an equation, expressing the equality 
of the quantities 12— -5 and 7. 

A aim/lie equation is that which contains only one un- 
known quantity, without including its power. 

Thus X— a+^ssc is a simple equation, containing only 
the unknown quantity x. 

Reduction if equations is the method of finding the va- 
lue of the unknown quantity, which is sho\m in the fol- 
lowing rules. 

RULE I. 

Any quantity may be transposed from one side of the 

equation to the other by changing its sign. 
Thus J i/'x-f 3=7, then will x=7— 3=s4« 
Andf if X — 4 -f 6 = 8, then will j? = 8 -f 4—6 a= 6.. 
Also^ (f x-^-^a+bsssC'-^dj'ihen will x=:c — t/-f c— d, 
Andf in like manner^ (/'4x— 8=3j:-f-20, then will 4.r 

.«.3x=:30-f 8, or a:=28. 

RULE II. 

If the unknown term be multiplied by any quantity, it 
may be taken away by dividing all the other terms of the 
equation by it. 



SIMPLE EQUATIONS. 77 



Tku8j ifax-ssab^-^y then will xas^^-— 1. 

jindj i/" 207+4=16, then wiil a:+2»Sj and xsbQ.^2 
= 6. 

In like manner^ if ax + 2ba = 3c», then 'mill x^2b b^. 

— • and x^ssi 2d. 

a a 



HULE III. 

If the unknown term be divided by any quantity, it may 
be taken away by multiplying all the other terms of the 
equation by it. 

Thusy (/•^=55+S, then wz7/ 07=10+6=16. 

X 

^4ndj if — =d+c— </, then wiU xssab+ac — ad. 
a 

2x 
In like manner j if 2=6+4, then will 2x — 6=18- 

36 
+ 12, a«c/2x= 18+ 12 + 6=36, or x^ — = 18. 



RULE IV. m 

The unknown quantity in any equation may be made 
free from surds, by transposing the rest of the terms by 
Rule I, and then involving each side to such a power as 
is denoted by the index of the surd. 

Thuay ifx/x — 2= 6, then will s/x = 6 + 2 = 8, and 
x=8a=64. 



^«rfi/\/4x + 16=12, then w«7/ 4a: + 16 = 144, or ^x 
= 144 — 16=128; and if both sides of the equation be 
divided by 4, x will 6e s= 32. 

G2 



ra SIMPLE EQUATIONS. 



In like manner^ «/-^2jr+3+4:5=8, then Vfill -^2. 
»8 — 4=4, and 2j7+3»43s64, and 2.r^64 — 3; 



61 
or x^T =30i. 



«|- 



2 '^^' 



RULE v. 



If that side of the equation which contains the 
known quantity be a complete power, it may be r^d 
by extracting the root of the said power from both 
of the equation. 

7%tt», «/x«+^x+9=25, then will x+3ses^25 = 

jindf r73:c«— 9=21+ 3, then will 3a?«=21+ 3 H 

33 
S$^an(/a?'s=---8ll, or xssy/\\. 
o 

In like manner^ if f- 10 = 20, then will 2j?2 

= 60, an(/x«+ 15 ass 30, or j:2±=30— 15 = 15, or 
V^15. 



RULE VI. 

Any analogy or proportion may be converted in 
equation, by making the product of the two mean f 
equal to that of the two extremes. 

7%M«, i/3x : 16 : : 5 2 10, then will Sx X 10 = 1( 

80 8 
or 3O:c=80, or x^ss — = — = 2|. 
' 30 3 ^ 

2 'P 2f ^ 

Andy if — :a:i b : C) then will — - sssab^ and 2cxs 
a li ■ 

Sab 
or xss ---. 
2r 



1 

I 

I 



SIMPLE EQUATIONS. 79 

In like manner jif 12— a? :— : : 4 : 1, /Ae« Wf// 12 — x^ 

^x 12 

■raa2x, and 2x+j:=a:18| or jrs=---=s4. 

RULE VII* 

^f the same quantity be found' on boAi sides of the 
^QUation with the same sign, it may be taken away from 
^fch ; and if every term in an equation be multiplied or 
<}ivided by the same quantityi it may be struck out of 
^^em all. / 

TAt/», if 4x+aaa:*+a, then will 4xssb, and x= 

b 



^f^ if 3oi:+5a*=B:8ac, then will Sx+56=8c, and 

"^ * 3 

2jr 8 16 8 
In like manner j if-^ ^ =— — --} then w/// 2x= 1 6, 

^nd xssB. 

MISCELLANEOUS EXAMPLES. 

1. Given Sx-^^\S^Bi2x+6 to find the value of or. 
Fir9tj 5a:— 3j:=b6+15, 

Orf 3x«=6+15«=21; 

SI ^ 
Jnd therefore xs«— =7. 

2. Given 40— 6j7— 16=120 — 14j: to find x. 
Fintj 14j>-6j:=s120 — 40+16, 

Or 8a7»136— 40bs96 ; 

96 
And therefore j?«r— a«12. 

8 



80 SIMPLE EQUATIONS. 

> • 

3. Let Sax — 3bs=i2da:+c be given to find x. 
Firat^ Sax — 2dx=iC+Zb, 

Or (5a — 2d)xx^c+3b. 

. c+3b 
And therefore 3?= 

4. Let 3j:2 — \Ox^=Bx+x^. be given to find x. 
Firsts 3x — 10=8+x. 

Or 3x — x=8H-io=l8. 

• 18 

And there/ore 2x=s\ 8^ or ar=--.s=9. 

Jt 

5. Given ^ax^ — Xlabx^-sizZax^Jc^^^S to find x. 
Firsts dividing the whole by Zax^^ we shall have^ 

Or 2x — x=2+4b. 
Whence x=2 + 4b. 

%J^ ^^ %A^ ^^^k 

6. Let r"+-r= ^^j ^^ given to find x. • 

2 3 4 

Firsts X 1 = 20. 

3 4 

Aho^ 3a— 2x-^ = 60, 

4 

^nt/ 12a: — 8jr+6x=:240. 

Therefore 10j7=^40. 

., ^ 240 

And x==-7— =24. 
10 

ir-"~~3 «>c X"^^ 19 

7. Given •— - +--=20 to find x. 

2 ^3 2 

2a 

i^e'r*?, X — 3H =40 — jr+19. 

3 

^/«o, 3ar — 9+2a:=!20— 3JC+57. 

Therefore^ 3a:+2a+oa-= 120+57+9. 

7%flrw, 8xxb|86, orx=— =23i. 



SIMPLE EQUATIONS. 8 1 



2j? ^ 

8. Let v^— +5aB:7} be given to finder. 

o 

FivBt^ ^— =7—5=2. 
' ^ 3 

Whence — =2^^^. 
3 

13 
jind 8x=a 1 2, or ^r:^---= 6. 

2 



2aa 



9. Let j?+v^fl»+xSaa . g , a be given to fiirdx. 

/Vr«f, J7^a »+xg+ a«+3:g=s2ag. 
Whence Xy/a^-^-x^^a^ — xS 

Or a^x'+x-^ssaa^ — ^2a«x»+x4, 

Or a*j7»-f-2a*j:2=sfl4, or3fl2j;a=a^; 

Consequently x^=s — ^ awrf x=!\/— =«%/—• 



EXAMPLES FOR PRACTICE. 

1. Given 3y— 2+24=31 to find y. Jna. y=3.* 

2. Given or+lSssSx — 5 to find x* Ana. ar=l 1^. 

3. Given'6— 207+10=20— 3x^2 to find X. 

jin^, ars=2. 

4. Given x+\x+^xcs\\ to find x. Ans, x=l§ 

5. Given 2j:— 4x+l=5x— 2 to find x. ^/z«. x=-. 

?• 

6. Given 3«x+— - — 3=djr— 4i to find x. 

^n9i x=-— -—•. 
6a— 2<!» 



82 SIMPLE EQUATIONS. 



7. Given ^x+^x — Jx=b-^ to find x. Ana. x 



I 
2 



.*, 



8. Given \^ 12-1- X =2+ v'J? to find jr. An%, x^ 

9. Given j?+a= to determine jt. Ana ,xss — 

a+x 



10. Given ^/a^+x^ssz(b^+x^)* to find x. 

Ans* j7M=v^ 



2a 



11. Given ^jc+^a-f-x-i to find x. 

12. Given 4- =^, to find x. 



13. Given a+x=v^a»-f-jc\/^+x^ to find a-. 



«i;{9. Xss — i— 
4a 



PROBLEM I. 



7 exterminate two unknown quantities^ or to reduce thet% 
simfile equations containing them to a single one, 

^ RULE I. 

1. Observe which of the unknown quantities is 
least involved) and find its value in each of the 
tions, by the methods already explained. 

2. Let the two values thus found be made equal 
ead^ other, and there will arise a new equation 
only one unknown quantity in it, whose value xnajl 
found as before. . 



♦•♦ 






SIM]|||^ EQUATIONS. 83 



£Xi;||j|L£S.* 



1. Given |^^+^^-^J^,tofind*andy. 
From, the first equation x= - 



2 
2 



10+2y 
And from the second x= — — ; 

5 

23— 3y 10+ 2y 
Consequently ■■■ ■ ^ = > 

<« d 

Or 115 — 15y=»20+4y, 

Or 19y=115 — 20=95; 

95 
That isy y= --- = 5, 
' ^ 19 ' 

23 — 15 
jfnd j7=— — =4. 

2 

2. Ghren ^ "^"^^^^ ? , to find x and f 

From the first equation jr=sa-— y, 

And from the second x^^b-\-y; 

Therefore ct^^y^ib-^y^ or 2y=ia — b ; 

a <o 
Consequently y=s , and x=a — y, 

_ a— -6 a-^b 

Or x^a = — — . 

^ 2 . 

5. Given ^t^tl^""!? , to find x and y. 

2y 
From the first equation ar=:?14— ~-, 

3 . • 

3v 
And from the second Xiss2^^-^—f^*^' 



rr. 






84 SIMPLE EQUA'qpNS. 

Therefore 14 — H£s=24— — , 

And 43 — ^^ys3iNi&— ^) 

Or 84— 4ysl44— 9y; 

Whence 5^=144— -84f^609 

60 
Andy^s^ — =12, 
^ 5 ' 

2y , 24 ^ 
3 3 

4. Given Atx+yss^^^ and 4^+^=169 to find x ai 

wfntf. ^ssBy and y 

^ ^,. 2j: 3y 9 .Zx 2y 61 ^ 

5. Given _ + _=_, and -+^ = 725. to fi. 

and y. ^ 1 j 

2 ^ 

6. Given x+ysssy andx*— y2=(/, to find ^ and y. 



i 



^ r ^. 2* ^ S 

7. Given X'^yssdy and ^ : y : : n : m^ to find x an 



RULE II. 



. 1. Consider which of the unknown quaflities ] 
would first exterminate, and let its value be found 
that equation where it is least involved. 

sNBubstiti^te the valub, thus found, for its equal 
the other equa^p, and there will arise a new equati 
with only oi^^lkknown quantitf, whose value may 
/bund a^b^brc. 



^^y.^ 



SIMPLE EQUATIONS. <S 

SXAMPJ.E8. 

*• Giwn J 3^^j; • JJ., to find X and y. 

-^rom the first equation^ j?s=sl7-— 2y. 

•'^nd ^At« value substituted for x in the second^ gives 

- (17— 25/)x|Ji-y=«, 

Or 51 — 6y — V5s=2, or 51— -7y=2 j 

7%a/ M, 7y =5 1 — 2=49 ; 

49 
Whence y=s — =7, anrf j?s5b17 — ^2y«17 — 14=3. 
7 

3. Given ?^^^^^3 ? > to find a? and y. 

jFVoto 4 he first equation jr=Bl3— j{. 

'^t' fAiff value being substituted f:>r x in the 2</y 

J»VM 13—1/ — y=:3, or 13 — 2ys:3, 

That is^ %= \ 3—3= 10, 

• »» 10 

' Whence ysa — =5,a72(/x=13— -y=13 — 5=S. 

'•Given J«;*^«;f;:^^, to find^^y. 

The first analogy turned into an equation^ 

oy 
ia bxssai/f or x=s -— , ^ 

<in£f /^ -va/ue qfx being subfttituted in the 2cf, 
Or ca|l+d3y2=fii2, or y^^-rrsi 




V 



86 SIMPLE EQUATIONS. 

4. Given 2x + 3y = 16, and 3a: — 2i/ = ll 
and t/. , jins* x=5j 

5. Given — 5|- 7y = 99,4pid — + 7^7 = 5 1 
and J/. •^/w. :r=7, ( 

6. Given— —12=-^+8, an^^ + — - 
+27, to find 3c and j/. ^/;z5. j:=60,* 

7. Given a : 6 : : x : y, and x^— t/S^s^, to fi 



nuLE in. 

1. Let the given equations be multiplied or 
such numbers or quantities as will make the t 
contains one of the unknown quantities the s; 
equations. ^^ 

2. Then, ^^fcding or subtracting the eq 
cording as is^requirtd, there will arise a ne 
with only one unknown quantity, as before. 



EXAMPLES* 

. Given p^JJ^^uJ.tofind^^andi 

Firhty viuUiply the 2d elation by 2 
Vfill give 3 J? + 6t/ =42. 

Then from the last equation subtra 

and it ivitl give 6i/— 5y = 42 — 40 

t ^nd therefore ;r sl4— 27/s:14-vi= 



n 

< 




SIMPLE EQUATIONS. %7 

2. Given J 3^75 p ^^ , io dad X and y. 

-Let the first equation be multifilied by 2, and the 2d by 
'^i and we shall have Ifi^"^ 6z/=5^ 

^fnd if the former of these be subtracted fro fn the latUr^ 

it will give iXysz&l^or ysB — =s2. 

fi 1 

9 + 3y 
^nd conieguentlyxsss — _, by the first equation^ 

5 

9 + 6 15 

Ancftht'r method, 

■Mukifily the first equation by 5, and the second by 3; 
and we shall have 5 ^"^"T, ^^^** ^ o 

^o«, /rr /Ae*f rwo equations be added together^ 

93 
and the%um will be 31j|p:93, or ^ses-— a=3. 



16 — 2x . mm 



And consequently^ y*= — — , by tXt^kond equation^ 

16 — 6 10 ^ 
Oryss s — asi2, as before. 



MISCELLANEOUS EXAMPLES. 

1. Give* ^^1^ + 8y==3l, and ^^ + 10x« 193, to 

3 4 

•lad JT and y. Ans. x=*\9j and ysn 3. 

2. Given — h 14=18, and _I— +16«l9,to 

find X and y. ^«. •■sS, and y 



88 SIMPLE EQUATIONS. 

. 3. Given — ~-i + ~ =:8, and -^-- — — j/=s 
find :r and z/. -^na, xa6, ancf 

4. Given ax + byss r, and djp + eyss/j to find x ; 

r.^— */• . a/ 
jfna. xsB — -, ana yss — 



PROBLEM II. 

2'o exterminate three unknown guantitiesy or to redu 
three aim/ile equations containing^ them to a single 

;_ RULE. 

1 . Let X, y, and z be the three unknown quantit 
be exterminated. 

2. Find the value of x from each of the three 
equations. 

3. Compardl|p fir&t value of x with the seconc 
an equation woRrise involving only y and z. 

4. In like manner, compare the first value of x 
the third) and another equation will arise involving 

y and z. • 

5. Find the values of y and z from these two equs 
according^ to the former rules, and Xy y, and z vv 

exterminated as required. ^ 

» 

M'ote, Any number of unknown quantities inay t 
terimnated in nearly the same manner; but ther 
often much shorter methods for performing the c 
tiooi which will be best ieamt from practice. 



rnXFLE EQVATtONS. t9 






^- Given -< jr-f2y-f 3za*62 J-, to fiiid j:, t/, wid z; 

i^r(W2 thejirst arssr 2 9— y—z. 

-fj'om Me second xssS^ — ^2y— 3z. 

^ 2z/ z 

From the third a:=20— .-^^— w; 

3 2 

Whence 29 — i^— Z3a62--.3y— ^z, 

2v z 

^^flf 29 — y — z=:20— — . 

^ 3 2 

Also from thefirnt of these ys=:33.i— 2r. 

3z 
And from the second yaB27— • — ^ 

Therefore 33 — 2zass27— — , or z^sil2r 

Whence also y=s33— -2z=9, 
Andx^29 — 9 — ^z=8. 

fi^+iy+i^«62-) Jt 
3. Given -< -J^+iy+l^— 47 V, to fuH jr, y, and z. 

^ir*r, the given equations^ cleared of fractions^ become 
12x+ 8y+ 6z= 1488 
20jc-f 15y+ 12z==2820 
30jr+24y-f20z=4560 

And^ if the second of these equations be subtracted from^ 
double the firsts and three times the tHrd from five Hmem, 
^ht secondy we shall have 

4x+ ys»l5« 
10x+3yaB42a 
H2 



oa SIMPLE JEQPATIOMS. 

And agatiij if the wond qf thewe be wbfrac^ed /r9i 
three ttmea thejfrat^it will give 

48 

fVr«/prtfy SB 156 -i-4x«: 60, (wrfr«sB — ■ g 

3. Given jr+y+zss53,a:+2y+3z=a 105, and jr-f- 
-(-4zsB 134, to find X| y^ apd z. 

^n«. a:s=24, vss6, ancf z=^ 

4. Givea jr?+ycitf| ar+mft, and y+zaasc, to find 
y, andz. 

f. Given < dx+ey+/z9sn v, to find Xj y, and z. 



A COLLBCTION OF ;|UESTIOMS PaODUCIN« 
SIMPLE EqUATXOHS. 



1. To find two numbers, such that their sum shall % 
40, and their difference 16. 

Let X denote the least of the two numbevM reqiured; 

Then will x-^- 16 as the gr eater f 

And x-\-x+ I6xs40, by the question^ 

That M, 3xa40— 16s24, 

24 
Or jrsB— >aBl3«3/«<r«^ number ^ 
3 

And x+ 16eBl2+ 16n283»5r«a^^ number requf 



SIMPLE EQUATIONS. 91 

2. What number is that whose ^ part exceeds its ^ 
part by 16? 

Lei x^Btnumbtr required; 
Then willits \fiart be —-, and tie ^ fiart — ; 

And therefore s=sl6, dy the question^ 

3 4 

That iSf x-^ — =48, or 4x-^3x^ 1 92 > 

XVhence xsas 192, the number required, 

3. Divide lOOOl. between A, B, and C, so that A stisdl 
^^\e 731 more than B, and C 1001. more than A. 

Let xsssh'a share qfthe given aum^ 

Then will x+72ssA'« »Aor^, 

j1ndx+l7^vaC*s3harey 

Mind the aum of all their aharea jc+-^+72+^+ 172i 

Or 307+244=1000, by the queation;. 

That ia^ 3:r=r 1000—344=756, 

756 
<9r a:=-l-=252/.=B'« ahare. 
3 

And x+72=252+72=384/.=A'* ahare. 

.^fiflrar+ 172=252+ 172=424/.=C*» ahare, 

252/. 

324/. 

424/. 



i 



lOOOl, the firoof, 

4. A prize of 10001, is to be divided between two per- 
sonsy whose shares therein are in the proportion of 7 to 
9 ; required the share of each. 

Let x^tfirat fieraon'a ahare ^ 
Then will lOOO'^'mX ^maecond fieraon* a ahare, 
And X ; 1000 — x iiT i9^ by the queationf 



92 SIMPLE EQUATIONS. 

That w, 9ar«.(1000— vc)x7«7000— rjF, 

Or 9x+7a7=16j:s=«7000, 

7000 

Whence x:=- ==43^/. \09,^Ut ehare^ 

16 

w^wdlOOO — :r=lOOO — 437/. 10«.=a563/. \Oa,^d share, 

5. The paving of a square, at 2s. a yard, cost as much 
as the inclosing it at 6s. a yard: required the aide of the 
square. 

Let x=s8ide of the square sought^ 
Then 4x=yards ofindosufe^ 
And x^sszyards of fiavement ; 
Whence 4xX5ssi20x=z/irice of inclodngy 
jln d^x^ X 2 = 3^2 s=/2 rice of paving; 
But yx^ssaaOjT, by the question^ 
Therefore 2x:=i20,- and xi:x\0=: length of the side re- 
quired, 

6. A labourer engaged to serve for .40 days upon 
these conditions, that for every day he worked he was 
to receive 20d. but for every day he played, or was 
absent, he was to forfeit 8d. ; now, at the end of the 
time he had to receive II. lis. 8d.; it is required to 
find how many days he worked, and how many he was 
idle. 

Let X be the number of days he worked. 
Then will 40 — x be the number of days he was idle^ 
Also xx20=:20j:=s«Mm earned,^ 
And (40 — jr)x8=S30 — ^xssisum forfeit edy 
Whence 20j7--(320— 8x)=380rf. (I/. 1 1«. 8c/.), by 
the question; that ia^ 20j? — 320+ SxsaSSO^ 
Or 2807=380+320=700, 

And ^==-r — =25=s7iMm^er of days he workvd^ 

And ^~^xaaa40<^26satlSjearnutnber of days he 
was idle. 



SIMPLE EQUATIONS. 93 

Out of a cask of wine, which had leaked away |, 3 r 
>ns were drawni and then, being guaged, it appeared 
I half full ; how much did it hold ? 

Let it be aufifiosed to have held x gaUons^ 

X 

Then it would have leaked — gallona^ 

X 

conaeguently there had been taken away2\ -{ — galiona. 

X X 

But 2 1 H isz~^y by the question^ 

3 i 

3x 
That isf 63+xas — , 

Or \^6+2xssSx. 
Hence 3a:— .2x=l26, 
Or x:^ 126= number ofgaUona required^ 



What fraction is that, to the numerator of whichi 
be added, the value will be ^ ; but) if 1 be added to 
denominator, its value will be ^ ? 

X 

Let the fraction be refireaented by ^» ; 

V 

Then vnll —I— = — , 

^nd = — , 

y+{ 4' 

Or 3x-|-3=y, 

^nd 4x«=y+l, 

Hence 4x— 3 j:— 3 = y + 1 ---y, 

Thatia^ x — 3=1, 

Or ar=s4, and yBB3x+3-il3+3=3l5, 

4 
So that — ^ss fraction required. 






94 SIMPLE EQUATIONS. 

9. A market-woman beught in a certain nnmber 
eggs at 2 a penny, and as many at 3 a penny, and s 
them all out again at the rate of 5 for two-pence, a 
by so doing, lost 4d. what number of cgjjs had she ? 

I^et xss number qfeggs of each nort^ 
Then will -^^szfirice (ifthcjirat aorty 

X 

And — as hrice of the second sort ; 
3 . 

But 5 : 2 : : 3x {the whole number of eggs) : • 

4uC 
Whence — firice of both aortsy at Jive /or 2d% 
5 

And 1- ^— — — a»4, by the question ; 

40 «i d 

^ . %x %x 

That i«, x-\ ==8 ; 

3 5 

24j7 
Or Sx+2x a«24; 

Or 15x+10ar — 24^=120; 
Whence xs=s 120= number of eggs of each sort require 

10. If A can do a piece of work alone in ten days, a 
B in thirteen ; set them both about it together, in wl 
time will it be finished ? 

Let the time sought be denoted by jt, 

X 

Then 10 days : 1 work : : x days : — , 

X 

And 13 days : 1 work : : xdays : •— ; 

X 

Hence — tssftart done by A in x daysy 

X 

And — ^sfiart done by B in x days; 

1 o 



'^S' 



SIMPLE EQUATIONS. 95 



X 






Consequently — H s=: 1; 

13jr 

, ' Thatis^ h xs= 13, or 13x+10:r= 130: 

10 . 

130 
And therffore 23x3sl30, or jc = -— = 5|^ daya^ the 

time required, 

11. If one agent, A« alone, can produce an effect e, 
in (he time a, and ariother a^cnt^ B. alone, in the time b ; 
in what time will they both together produce the same 
effect? 

Let the time nought be denoted by x. 

ex 
Then ai e i i x i — =^iiart of the effect firoduced by A, 



tt 

ex 
.-M, bi e I I X I ^^ fiart of the effect firoduced by B; 



ex t'^ 
Whence 1 =: e^ by the question ; 

a ^ 

^ X X 

Or--4--=i; 
a b 

That i«, xA =a ; 

b 

Or bx-^ax^ssba ; 
And consequently x=. == time required. 



Ql^ESTIONs FOR PRACTICE. 

*• What two numbf:rs art those, whose difference is 7. 

^'^^l sum 33 ? Ans. 1 3 and 20. 

^» ^« To (livide the number 75 into two such parts, that 
• 5^e times the greater may exceed seven times the less 

^* 15. Ans» 51 and 21, 



96 SIMPLE EQUATIONS. 

3. In a mixture of wine and cfder* ^ of the whole /i/im 
25 gallons was wine, and ^ part minua 5 gallons was cjr-^ 
der ; how many gallons were there of each f 

jirm. 85 qfwiney and 35 of cyder ^ 

4. A bill of 1201. was paid in guineas and moidore^. 
and the number of pieces of both sorts that were use^^ 
was just 100; how many were there of each? 

Jna, 50 ofeac^^ 

5. Two travellers set out at the same time from Lon- 
don and York, whose distance is 150 miles; one of then 
goes 8 miles a day, and the other 7 ; in what time wiU 
they meet ? ^na, in 10 dayv^ 

6. At a certain election, 375 persons voted, and th^ 
candidate chosen had a majority ol 9 1 ; how many voted 
for each? Ana. 233/or une^ and \42/br the othewr'^ 

7. What number is that from which, if 5 be rot>* 
tracted, f of the remainder will be 40 ? jina. 6S« 

8. A post is \ in the mud, ^ in the water, and 10 feet 

above the water \ what is its whole length ? 

jina* 24t Jeet, 

9. There is a fish whose tail weighs 91b. his head 
weighs as much as his tail and half his body^ and'his 
body' weighs as much as 'his head and his tail ; what is 
the whole weight of the fish? Ana.T^it. I| 

10. After paying away i and | of my money, I tad 66 
|;uineas left in my purse ; what was in it at first ? 

Ana* ]20 guineai* 

11. A's age is double of B% and B's is triple of C% 
and the sum of all their ages is 140; what is the age of 
each? Ana. A'*=«=84, B'«=42, and C'4««i4 

12. Two persons, A and B, lay out equal sunu of 
money in trade ; A gains 1261. and B loses b71. and A'a 
money is now double of B's; what did each lay out? 

Ana. 3Mi 



I 



> 



SIMPLE EQUATIONS.'^ 97 

13. A person bought a chaise, horsei and harness, for 
^. the horse came to twice the price of the harness, and 
^'le chaise to twice the price of the horse and harness ; 
^hat did he give for each ? 

jins, 131,68, 8d,Jbr the horse^ 61, ISa, Aid, for the 

harness^ and 40i,/br the chaiae, 

14. Two persons, A and B, have both the same income : 
^ saves -J of his yearly, but B, by spending 501. fier an- i 
^Um more than A, at the end of 4 years finds himself 
'OOl in debt: what is their income ? »/ina, 125/. 

15. A person has two horses, and a saddle worth 501. 
^f)W if the saddle be put on the back of the first horse, it . 
*^11 make his value double that of the s&cond ; but if it 
"^ put on the back of the second, it will make bis value 
Uiple that of the first ; what is the value of each horse ? 

^w«. One 301, and the other 40/. 

16. To divide the number 36 into three such parts that 
•} of the first, \ of the second, and ^ of the third, may be 
ail equal to each other, ,/^na. The fiarta are Qy l^^and i6. 

17. A footman agreed to serve his master for 81. a year 
and a livery, but was turned away at the end of 7 months, 
and received only 21. ISs. 4d. and his livery ; what was itH 
value ? >!fna, 4/« 16^. 

18. A person was desirous of giving 3d. apiece to some 
beggars, but found he had not money enough in his poc- 
ket by 8d. be therefore gave them each 2d. and had then 
3d. remaining ; required the number of beggars, j^na, 1 1 . 

19. A hare is 50 leaps before a greyhound, and takes 4 
leaps to the grcyliound's 3 ; but two of the greyhound's 
leaps are as much as three of the hare*s ; how many leaps 
must the greyhound take to catch the hare ? Jna, 300. 



98 SIMPLE EQUATIONS. 

20. A person in play lost A of his money, and then woi 
r> shillings ; after which he lost | of what he then ha( 
and then won 2 shillings; lastly he lost ^ of what he thei 
had; and, this done, found he had but 12s. remaining 
what had he at first ? jins, 20a 

21. To divide the number 9CV into four such parts 
that if the first be increased by 2, the second diminished b] 
2, the third multiplied by 2, and the fourth divided by 2* 
the sum, difference, product, and quotient shall be a] 
equal to each other. 

jlna. The parts are 18, 22, 10, and 40, reafiectivel^ 

22. The hour and minute hand of a clock are exactT 
together at 12 o'clock; when are they next together? 

jlna. I ho. 5r^ nti 

S3. A man and his wife usually drank out a cask < 
beer in 12 days; but when the man was from home, 
lasted the woman 30 days ; how many days would the mjE 
alone be in drinking it f .^ns, 20 dai^ 

24. If A and B together can perform a piece of wor 
in 8 days ; A and C together in 9 days ; and B and i 
in 10 ^ys'f how many days will it take each person t< 
perform the same work alone ? 

Jna. A 14-11 daysj B iT^^and C 23/j. 

25. If three agents, A, B, and C, can produce the ef- 
fects a, A, c, in the times e, /, g^ respectively ; in ivhat 
time would they jointly pix)duce the effect d? 



V 



[ 99 ] 



QUADRATIC EQUATIONS. 

•^ sim/ile quadratic tquation is that which involves the 
*^Uare of the unki>own qiiaiitity only. 

•An adjected quadratic equation is that which involves 
tJie square of the unl;.nown quantity, together with the 
product that arises from multiplying it by some known 
quantity. 

T'huB ax^ = 3, is a simfile quadratic equation^ 
-^nd ax^-j- bx = c, is an adfected quadratic equation. 
The rule for a simple quadratic equation has been given 
already. 

^ All adfected quadratic equations fall under the three 
'^Mowing forms : 

1. a72«j_ axsszb 

2. x^ — ax=s b 

3. x'^-—^ax^=i-^b. 

The rule for finding the value of x^ in each of these 
^^uatioDs, is as follows: 

RULE*. 

1. Transpose all the terms which involve the unknown 
quantity to one side of the equation, and the known terms 
to the other, and let them be ranged according to their di- 
mensions. 



• The square root of iny quantity may be either-j-or — , and there- 
fore all quadratic equations admit of two solutions. Thus the square 
root of -f-'i^is -j-M or — n; for (+«) X (+'^) o' ( — ") X ( — ^) arc 
each equal to -{-«2, but the square root of — «2 or ^ — «2, is ima- 
ginary or impossible. 



;v,v,vAv;\\ 



100 QUADRATIC EQUATIONS. 

2. When the square of the unknown quantity has an; 
co-efficient prefixed to it, let all the rest of the terms b 
divided by that co-efficient. 

3. Add the square of half the co- efficient of the secon< 
term to both sides of the equatioUi and that side whicl 
involves the unknown quantity will then be a complet 
square* 

4. Extract the square root of both sides of the cqua 
tion, and tlie value of the unknown quantity will be de 
termined, as was required. 

J^ote 1. The square root of the first side of the equa 
tion is always equal to the unknown quantity, with ha] 
the co-efficient of the second term subjoined to it • 

2. All equations, in which there are two tenms in 
volving the unknown quantity, and which have the inde: 
of the one just double that of the other, are solved, lik 
quadratics, by completing the square. 

Thus^ x^ + 0x2= d, or x^^ + ax«== d, are the' same a 
quadratics^ and thf, value of the unknown Quantity may b 
determintd accordingly. 



since either of them, being multiphed by itself, will produce ^-f-^oS 
And this ambiguity is expressed by writing the uncertain sign J 
before y/ib+ia^)-, thus Jf=±v'(^+ia»)-- i^- 

In this form, where x:am±.^(b-\-la^) — \aj the first value o 
X, viz. x=-f.^(A — \a^) — -Ja is always affirmative; for sine 
^€fi-\-b is greater than ^^2, the greatest square must necessarily hav^ 
the greatest square root; therefore ^/(^b-\-^a^) will always be great 
er than v^ia^, or its equal la; and consequently -}-^(5_j_^ii2^ — ic 
will always be affirmative. 



I 



QUADRATIC EQUATIONS. 101 



EXA.MPLES. 



1. Given j72+4x=i40, to find x. 

first J ar3+4x+4= 140+ 4=144, by completing the 
*9Uare, 

Then v^(j?«+4j?+4) = v^144, by extracting the root; 

Or, which is the same thing, :c+2=s 12, 

^ JInd therefore x^=\2'-^2^=^\0. 

^. Given x^ — 6jf+8=80, to find x. 

First, x^ — 6^7=80 — 8=72, by transposition ^ 
/^en x^ — 6^+9=72+9=81, by completing the square, 
Jlnd JT— 3=-v/81=9, by extracting the root ; 
Therefore x=9 + 3=l2. 



Tbe second value, viz.x=s— y/(^+^a*) — ^a, will always be^ne- 
itive, because it is composed of two negative terms. Therefore, 
"^Vhen x2+ax=A, we shall have x=+^(A+da3) — ic for the 
Affirmative value of x, and xz=s. — \/{b-\-~a^) — la for the negative 
"Value of X. 

In the second form, where x=s-^y/(b-^la^)-{-iay the first value, 
Viz. x=+y/(^+ a2)-|_^a, is always affirmative, since it is com- 
'^osed of two affirmative terms. The second value, viz. x= — y/(b 
^^a2)+ia will always be negative; for, since A+i^a is greater 
than ^a», \/{b-\--^a^) will be greater than ^-fl2, or its equal ^a; 
^nd consequently — \/(^~t--l^^)+i^ ^^ always a negative quantity. 

Therefore, when x2 — a.v=^, we shall have x-=z-\.^/(b-\-^.a^)-^ 
-Jfl for the affirmative value of x, andx=: — ^/(J>-\- a'i)-^.a for the 
negative value of x; so that, in boih the Hrsv und second forms, the 
unknown quantity has always two values, one of which is positive^ 
and the other negative. 

1 2 



i -^if 



lOS QUADRATIC EQUATIONS. 

3. Given 2a:2+8j? — 20=70, to find x. ' 

Firstj 2x24-8ar=70+20=90, 6y transfiosUion. 
"'! L Then x^+4x=s^5y by dividing by 2, 

jind :r2+4x+4=495 by comfileting the square ; 
Whence x+2:s:iy/4i9^=7^ by extracting the root, 
^nd consequently x==7 — 2=5. 

4. Given 3x^ — 3.r+6=5^, to find x, 

"^ Here x^ — :r+2 = l^, by dividing by 3, 
^nd x2— a:ss: 1^ — 2, by transjiosltion ; 

Also x^ — jc+^= l7..«.2-f. 1 -s-^^j by comfileting the squc 
And X — J=^yig-=J, by evolution ; 
Therefore j:=J-|-^=7) the answer. 

5. Givfen ^-— ^ +20A=42|, to find x. 

x^ X 
Here --— — -=42| — 20A=22|., by transfiosition^ 

2x 
And x2— — =44.|, by muUi/ilying by 2. 

Whence x^ f-i=44^4-^=44|, by comfileting 

squarcy 

And X'~^ssiy/4f4^-^6^y by evolution ; 
There/ore a; was 6^ -4-^=7, the answer^ 



In the third form, where x=sdts/(icfi — ^)-j-Ja, both the vs 
of X VfiW be positive, supposing la^ h greater than b. For the 
"'. vvalue, viz. x=-f-^(ifl2 — 6)+^a will then be affirmative, b 
composed of two affirmative terms. 

The second value, viz. x= — y/Qa^ — b)-\-ia is affirmative! 
since ^a^ is greater than r^fl*— J, y/^fl*, or la, is greater 
\/(id^ — b)i and consequently — ^(^a2_«^)«|_jflf will. always b 
affirmative quantity. Therefore, when x^ — axssz — b, we shall ! 
.v=+v^(ia3— Z»)4.^cf, and also x=— v^(ia2-H^)+ffl, for 
affirmative values of x. 



^ 



QUADRATIC EQUATIONS. 10 



A 



^» Given flx*+dx=:c, to find x> 

r» be 

^irst^ x^-^ xs= — , by division; 

a a 

^ b b^ c b^ 

rn, x^-\ j7 -J ss 1 by comfileting the square. 

a 4a* a 4a* « 

^ b f c b^\ /4ac+d2\ , 

• Given ax^ — te+c=rf, to find x. 
•Herey ax^~~-'^xs=sd.,^^Cf by trans/iositioriy 

Jlnd x2— — x= , by division, - 

a a 

b b^ c?— c b^ 

^ x^ X'\ =— 1 ^by comfileting the square^ 

a 4a2 a 4a^ 



b Id — c b^ \ 

And ^--±V ^-^+ ^,j, by evolution ; 

Therefore x^ ±±y/ f — +— V 

^. Given x^+2ax^sstby to find x. 

^ej x^-i-2ax^'\'a^=::b+a^y by comfileting the square^ 

And x2+.a=s^(d4.a2), by evolution ; 

Whence :c2«v^(<5+a2>--a^ " 

And consequently j?=V — a+y/^b-^-a^). 



But in this third form, if ^ be greater than Jfl«, the solution of 
i proposed question will be impossible. For since the square of any 
intity(whether that quantity be a^rmative or negative) is always 
Tmative, the square root of a negative quantity is impossible, and 
mot be assigned. But if ^ be greater than ^fl», then -40% — b is a 
(ative quantity; and therefore ^(JfC^ — b) is impossible, or ima- 
lary ; consequently \tk that case xssiiet±iy/{\a^^b) is always 
possible, or imaginary. 



104 QUADRATIC EQUATIONS. 

n 

9. Given aj?" — bx^ — c=— </, to find x, 

n 

Firsts ax^ — bx^^sszc—^^ by iranafioaitiorij 

b n c d 

jind x^ x^=s , by division^ 

a a 

d n 1,2 C-— e/ b^ 
JUo x" ;c^H = f-— T, by com/ileting t 

square, 

n I, /c— ^ b^\ , 
And x^ xsik/ I h — :; ) by evolution ; 

n b /C — d b^ \ 
^wa consequently x= [ — ±v^^ ^ J n • 

EXAMPLES FOR PRACTICE. 

J. Given x2 — 8jr+10=sl9, to find jc. Ans,x=^ 

2. Given x^-^-x — 40=170, to find x. Ana, x=l 

3. Given Sx^+Sx — 9=76, to find x. Ana. x=^ 

4. Given ^x^ — Jx+7|=8, to find x» Ana, x=s i 
6. Given 2x^ — x2=496, to find x. Ana. xsss 

6. Given 4^ — Jv^x=22|, to find or. «^n«. ^=4! 

7. Given |ji;2+3^==|, to find x. ^n«. .6685 

8. Given 07^+6x3=2, to find j:. . 

Ana. x^^—S+s/U 
9.' Given x^+x=ay to find x. Ans.xs=\/(^a+\) — i 

10. Given j?— v^jr=o, to find x. _ 

Ana. x=(i+V^«+iJ 

1 1 . Given 3x^n — 2*'»=s25, to find x. 

-^na. :r=(^v^76+^ 

12. Given \/\+x — 2^^l.fj?=4, to find x. 

Ana. x=Cl+-v/5>— 



QUADRATIC EQUATIONS. 105 



f 



QUESTIONS PRODUCING QUADRATIC EQUATIONS. 

I. To find two numbers whose difference is 8, and 

product 240. 

• 

Let x=Bthe leas number i 
Then will x+Sssthe great er^ 
^nd xXi^+S)=x^+Sxss24iOy by the question; 
Whence j??+8j74- 16=240+ 16=256, by completing the 
square, 

jiUo jr4-4=\/256=16, by evolution; 
'^nd therefore j:=16 — 4=:12aB/f5« number ^ and 12 + 
^^20ss greater* 

3* To divide the number 60 into two such parts that 
their product may be 864. 

Let x=sgr eater fiart^ 
Then will 60— x=/eM, 
4nd xX(60— a.)=60j:— J72--864, by the question; 
That M, x^ — 60j:= — 864 ; 
^^hence or*— 60x+900= — 864+900=36, by completing 
the square. 
Also X — 30=v'36=6, by extracting the root; 
Jnd therfifovi? x=6+S0=x36^ greater fiart) 
And 60 — a?=60 — 36=24 = /<?««. 

3. Given the sum of two numbers =10 (a), and the 
sum of their squares = 58 (6), to find those numbers. 

Let Xiasgrjater of the two numbers; 

Then will a— x= less; 

And jr*+(a— x)2=2a'2+aa — 2cix=d, by the question^ 

a* b 

Or x*+ — — ax^ — J by division. 



106 QUADRATIC EQUATIONS. 

b a^ Ih'^a^ 
Or X*— cj?=» — — — =ssr. , by iransfiosi 



a2 ^«_a2 q2 2 b — a^ 

IVkence x^— ^cj: +-^ = — 7: 1 = > ^y 

4 2 4 4 

ing the square, 

a 2b-'-~a^ 
Mao X s= ^ , by extracting the 

<6 4 



2b a2 a 

And therefore a?=±v/ 1 ^= greater n 



. . (a ^2d— an ^2d— fl2 a 

Hence these two theorems^ being fiut into numi 
7 and 3 /or the numbers required* 

4. Sold a piece of cloth for 241. and gained 
fier cent, as the cloth cost me ; what was the pr 
cloth ? 

JLet xsssfiounda the cloth coat; 
Then 24i^-x=swhole gain; 
• £ut 100 : X I : X : 24— a:, dy the que 
Or x2=100X(24 — x)=2400— IOOjt; 
That ia x^+ 100ar=2400 ; 
Whence x^+ iOOa:+2500=2400+2500=490C 
fileting the aguare. 
And j:'+50=:-v/4900==70, by extraction of n 
Consequently xssx7O--~^50=:20l=:/irice of the 

5. A person bought a number of oxen for 
if he had bought four more for the same n 
would have paid U. less for each; how mai 
buy? 

Let the number of oxen be refireaented I 

80 
Then will — be the fir ice of each. 

X 



QUADRATIC EQUATIONS. 107 

80 
-^nd —r:=^f^^^^ ofeachy t/s:+4 had coat 80/. 

^ 80 80 
^ut — = — —- +lj by the question* 

^ 80x 

Or 80= \-x. 

x+4^ * 

Or 80x+320=80j7+a:*+4x, 

••^ce ara + 4j;+4= 320 + 4 = 324, by comJUeting tife 
^quaref 

And j?4-2=\/324s=b18, by evolution; 
^^equently x=i 18 — ^2=16 == number of oxen required. 

r 6. What two numbers are those whose sum, produ(^, 
'ana difference of their squares are all equal to each 
other ? 

Let a:= greater number^ 
And yzszleaa; 

And 1 = — --— ssx — y, or arx=y+ Ij/rom the 2d equation; 
x-f-y 

Also (y+ l)+y=(y+ l)Xyj from thefirat equ^iion^ 
Or 2y+l=sy2-^lyj 

7%a^w,9«-y=l; 
Whence y*— y+^^l^j ^y completing the aquare; 

1 5 %^5 

^/so y— — =^1^=:^ — s — -, by evolution i 

Conaequently y = -— H = -— — , and j?=s y + 1 ass 

^- — 3I_. And if these exfireaaiona be turned into nuwbensy 
%ve ahall have a:s=2.6180+ 

fl7Zrfl/5rsl.6l80 + 



108 QUADRATIC EQUATIONS. 

7. There are four numbers in arithmetical progr^ 
sion, of which the product of the two extremes is 45| 
that of the means 77; what are the numbers? 

JL,et x^s^lesB extreme, 
and y s=scommon difference; 

Then x^x+yj x+2yj :r+3y, will be the four nimbert 

«"" ^(x+y)x(a7+2y)=^*+3ary+2ya=775 gue9H(M^^ 

32 

Whence 2y2--s 77—45 =32, by subtraction^ and y^Ga-^^s^ 

16, by division, * 

Or yss^l6=s4, by evolution; _ 
Therefore x2+3xy«»:jp*+l2ar=45, by the Xst equation; " 
Mao 072+12^7+36=55 45+36=81, by completing the^ 
square^ 

And j:+6=^81=s9, hy the extraction of roots; 
Consequently x=9— *6s=3, and the numbers are 3, 7, llj^ 
and 15. 

8. To find three numbers in geometrical progression^ 
whose sum shall be 14, and the sum of their squares^ 
84. 

Let X, T/, and z be the numbers sought; 
Then xz=sy^y by the nature of firo/iortitmj 

•^'^^ \ ^2+y2+23^ Jl \ > *y '^^ quesHon; 

But j7+z=14— y, by the 2d equation* 
Jnd ar2+2j7Z+z2= 1 96— 28y +^2^ by squaring both sidei^ 
Or x2+22+2i/»=196— 28y+y2, byfiutting^y^.for itff 
equal 2xz; 
That is^ x2+z«+y2=196— 28y, by subtraction^ 
Or 196— 28y=84, by equality; 

196 — 84 
Hence yss _ =s4, by transfiositton and division. 



/ 



QUADRATIC EQUATIONS. 4^9 

16 
fgairij a:zs=y2^16, or ar== — by (he \at equaiion, 

z 

16 

'^ndx-^-y-^-z^n ^- 4 + z = 14 by the 2d equation^ 

z « 

r^r 16+42 + z2=r 14Z, or Z^ lOZras — 16, 

IVhence z^ — lOz +25= 25 — 16 = 9, by completing the 

iguare^ 
^nd 2 — 5 = v^9=3, orz = 3 + 5=8, 
Consequently^ x = 14 — y — z = 14 — 4 — 8 ==2, and 

the numbers are 2, 4, 8. 
9. The sum («) and the product {fi) of any two num- 
^^^rs being given; to find the sum of the squares, cubes, 
^^cjuadrates, &c. of those numbers* 

Let the two numben be denoted by x andy. 

Then will i "^ •" ""* *> by the question. 

But (j:+y)*=x*+2jpy+y*=«*, by involution^ 
and x^'{'2xy'\-y^''^2xy=s8^ — 2/2, by subtraction. 
That is^ j7«-f-y25— «2 — 2fi9=i8nm o/ the squares. 

-^gainj (a:2+y2)X(a?+t/)=(«2— 2/2)X«, by multiJiUcation^ 
Or J734-j:yx(x+y)+y3=fi3 — 2*/i, 

Or x3^«^+y33--«3 — 2»/», by substituting .72 for its 

equal 3cyx{x+y)\ ./^^ 
And therefore x^+t/SssaS — lJ^/iaBssttm q/*/^? cubes* 
-in tike manner^ (x3-fy3)x(j?+i/)=(«3 — 3sfi)xsy by mul" 
ti/iUcation^ 

Or ar*+ary X(j?2+y2)+y4=:«4— 3«Vi, 

Or x^+/iX(«* — 3/i)+y*=«'* — 3*% by substituting fix, 

{«« — 2fi) for its equal xyx{x^+ y*) i 
Andt conset/uentlyy x^+y* aesi*— 3«V*'— /^ X («2 — 2/j) »=*<— 

4s^fiJ^2fi^ ssi sum of the biquadrates^ or fourth powers ; 
And the sum of the nth, powers is 4" — ni^^-^p -f x . 
«— 3 ♦»•— 4 , « — 4 n — 5 w — 6^, , n — 5 

n— .6 n — 7 w — 8 
1 J-.* M^c. ^ 



no QUADRATIC EQUATIONS. 

to. The sum (a) and the sum of the squares (6) 
four numbers in geometrical progression being give 
to find those numl^rs. ^ 

Let X and y denote the two means. 

Then will — and ^— 6e the tivo extremes^ by the nati 
y X 

offirofiorlion. 
MsOy let the sum of the two means = «, and their fi 

duct =fi^ 
Then will the sum of the two extremes = a — fS by 
question^ and their product =s/ij by the nature qffi 
portion, 

fx^ + y^ ^ %., ij^ ^ 1 
. Jitnce} x^ y^ ^^^ ^y Uy the iastfirobla 

Jnd x^ +y^ + ^ +^-«* + (a~«)2— 4/i=6, by 

y x^ 

question. 

x^ y^ 

Again^ 1 = c— », by the question, 

y X 

Orx3+i/=J?t/X(a— «)=//X(fl— «)• 
But j:^-4-J/^==*' — 3A/i, by the last fir oblem; 
And therefore fiX{a — *r)=A3 — 3«/?, by equality^ 
Qr pa — pH+:itis^pa-\''Zps^=^is^y 

Or p=s by division; 

- a+2» 

Whence «2+(a— «)2— 4/i=&a+Ca—«)2— — — =i, 

substitution, "^ 

b a^-^b 

Or s^"\ »= f by reduction, 

a 2 

/a* — b ** \ * 1 

and extracting the rooi, 
Andffrtm this value qftj aU the rest of the quantities 
Xy mnd y^^inay be readily determined. 






QUADRATIC EQUATIONS. 1 1 1 



QUESTIONS FOR PRACTICE. 

1. What two numbers are those whose si^tn is 20, and 
*^>eir product 36? Ans, 2 and \%, 

2. To divide the number 60 into two such parts, that 
*^eir product may be to the sum of their squares in. the 
^atio of 2 to 5. Ana. 20 and 40. 

3. The difference of two numbers is 3, and the diffe- 
^nce of their cubes i^ 1 1 7 ; what are those numbers ? 

4. Ana. 2 and 5. 

4. A company at a tav^nr hia^I. 153. to pay for their 
reckoning ; but, before fc'bU^ivii settled, two of them 
left the room, and then llrtirii'rwrtwjri iinini il had 10s. a- 
piece more to pay than JWIblBif iiow many were there 
in company ? ^ ^"^ Ana, 7. 

5. A grazier bought as many sheep as cost him 601. 
and, afiei; reserving 15 out of the number, he sold the 
remainder for 541. and gained 2s. a head by them; how 
many sheep did he buy ? Am, 75, 

6. There are two numbers whose difference is 15, 
and half their product is equal to the cube of the lesser 
number; what are those numbers? Ans, 3 and 18. 

7. A person bought cloth for 331. 15s. which he sold 
again at 21. 8s. fier piece, and gained by the bargain as 
much as one piece cost him; required the number. of 
pieces. Ans. 15. 

8. What number is that, whicn, when divided by the 
product of its two di{>;it3, the quotient is 3; and, if 18 
be added to it, the digits will be inverted? Ans, 24. 

9. What two numbers are those whose sum, multi- 
plied by the greater, is equal to 77; and whose diffe- 
rencC) multiplied by the lesser, is equal to 13 ? 

Ans, 4 and 7, or f^/S amd V \/3 



>4r' % 



1 12 QUADRATIC EQUATIONS. 

10. To find a number such that if you subtract it 
from 10, and multiply the remainder by the number 
itself, the product shall be 21. ^na, 7 or 3. 

11. To divide 100 into two such parts, that the sum 
of their square roots may be 14. 

jlns, 64 and 36. 

12. It is required to divide the number 24 into two 
such parts, that their product may be equal to 35 times 
their difference. -^ns, 10 and 14. 



/■-> 



13. The sum of t^o numbers ia 8, and the sum' of 
their cubes is 152 j what are the numbers ? 

^ns, 3 and 5. 

14. The sum of two numbers is 7, and the sum of 
their 4th powers is 641$ vhat tfre the numbers? 

.' ' Ma, 2 and 5. 

15. The sum of two numbers is 6, and the sum of 
their 5th powers is 1056; what are the numbers? 

jina. 2 and 4. 

16. The sum of four numbers in arithmetical pro- 
gression is 56, and the sum of their squares is 864; 
what are the numbers? ^na, 8, 12, 16, and 20. 

17. To find four numbers in geometrical progression 
whcsc sum is 15, and the sum of their squares 85. 

ylua. 1, 2, 4, and 8. 

18. It is required to find four numbers in arithmetical 
progression, such that their common difference may be 
4, and their continued product 176985. 

^ jina. 15, 19, 23, and 27. 

19. Two partners, A and B, gained 1401. by trade; 
A's money vvas three months in trade, and his gain was 
GOl. less than. his stock; and B's money, which was 501. 
more than A's, was in trade five months;' what was A*s 
stock? ji7i8. 100|. 



% * 



[ 113 3 



•F THE NATURE AND FORMATION OF 

EQUATIONS IN GENERAL, 



■ 

All equations of superior orders are generated by the 
moltiplication of those of inferior orders, involving the 
same tinknown quantity. 

Thus, a quadratic equation is formed by the inuUipll'- 
cation of two simple equationt. 

A cubic equation is produced by the continued multi- 
plication of three simple equatioMf or by one quadratic 
and one simple equation. 

A biquadratic equation is generated by the continued 
multiplication of four simple equations ; or by two qua- 
dratic equations ; or by one cubic and one simple equa* 
tioii) See. 

For, suppose the unknown quantity to be x, and its 
values in any simple equation to be a, ^, r, </, &c. 

Then those simple equations^ by bringing all the terms 
to one side, will become x-— aaeso, x-— d^o, x — c 
^ 0, X— d =s 0, 8cc, 

And the product of any two of these, as (x-~a) x 
(x— 6)b8so, will form a quadratic equation j or one of 
two dimensions. 

The product of any three of them, as (x — c) x 
(x— 6) X (x— c) = 0, will form a cubic equation^ or one 
•f three dimensions. 

r The product of any four of them, as (x — n) x (x— b) 
X (x— - c) X (x— >(/) ass 0, will form a biquadratic egua- 
tion^ or one of four dimensions, &c. 

From hence it appears, that every equation has as 
many roets as it has simple equations which produce^ 



I 



I i i K ATURE OP EQUATIONS. 

it, or as there are units in the highest dimension of the 
unknown quantity. 

For, if any of the values of x (a, b, c, d) be sub- 
stituted in the place of x in the biquadratic equation 
(r — a) X (x— 6) X (jc — c) X (x— rf), all the terms of. 
that equation will vanish, and the whole will be equal to 
nothing. 

And, as there are no other quantities, besides these 
four, which, substituted in the place of x, will n\ake the 
|>roduct vanish, it is plain tliat the equation cannot pos- 
sibly have more than four roots, or admit of more than 
four solutions. 

After the same manner, it may also be shown that no 
equation whatever can have looie roots than it contains 
dimensions of the unknown quaolity. 

To make this still plainer by an example in numbers ; 
suppose the equation to be resolved be x^ — IOjt^ -|- 
35jp2_50j7 + 24 = 0, and that you discover this equ«» 
tion to be the same with the product of (x— -l) x (x— 2) 

X(x— 3)X(:c — 4). 

Then it may be inferred, that the four values of x are 
1, 2, 3, and 4; for any of these numbers being put for 
X will make that product, and consequently x^ — \0x^ 
+ 35j?2_50x + 24, equal to nothing, as it is in the pro- 
poaed equation. * 

And it is certain that there can be no other values of 
X besides these four ; since, if any other number bt; sub- 
stituted for X in those factors, none of them will vaaish, 
$fid therefore their product cannot bo equal to nothing, 
as it ought to be by the equation. 

The roots of ec^tions are, also, either fioeiUve or 
vegatrvey ftccording as the roots of the sin>p]e eqtiatiofi 
from whence they are produced are positive or negative. 

Thud, if you suppose a: =— a, xsxsdy xms — c, and 
xaa df Uien will x+avmoj x^^d^so^ x+Cf^Q^ wad 47-— d 



> , 



-i. 



NATURE OF EQUATIONS. 



115 



=0, and Ihe equation (x+a) x (jc— ^) X (^+c)x{x-^d) 
BO9 will have its r«ots — a, +dy — c, +d» 

But the »igne and co-efficienia of equations will b^ best 
inderstood by consideiMiig the following table; where 
he: si lYiple equations X — a, x — 6, Sec. being multipiitd 
UMiiinuaily togtiber, produce, successively, the higher 
:qualions. 



X*— ax 






X X— ^=0 



— d Ix^-j-^^ VX— tf^CassO, a cubic, 



Xj'^-^ssO 



• — ar\ 






abc^ 
bd\^x '{' abcdssz 0, a biguadna* 



>x^ 



^■^acd \ 



tlCy 



bed) 



&c. 

l^^rom the inspection of these equations it is plains tHat 
^lie co-efficient of the first terms is unity. 

The co-efficient of the second term is the 9um of dU 
^ht TQfkU (a, A, c, rf) vfith contrary eigns* 

The coefficient of the third term is egual to the sum 
^tfihe reetangiea qfthe roots^ or of\>U the firoducta that 
^«n Jioaaibly ame by combriirff them tvfo and two. 



lie NATURE OF EQUATIONS. 

The co-efficient of the fourth term is equal to t 
of all the firoducts that can /losaibly arise by con 
them^ three by three ; and so on for any other co-e 
whatever. 

The last term is always equal to the firoduct oj 
roots with contrary signs; and this reasoning wi 
"Whether the roots be positive or negative. 

It likewise appears, from inspection, that the s 
all the terms of any equation in the table are alte: 
-f and — . 

Thus the first term is always some pure powe: 
an^is positive. 

The second term is some power of x, multipl 
the quantities — 'a* -— ^, — r, &c. and, since these 
titles are all negative, it follows that the term itsel 
be negative also. 

The third term has the product of any two of 
quantities ( — a, — d, — c) for its co-efficient^ J 
therefore positive ; since — x — , as well as + X +j 
-f-, or an affirmative quantity. 

For the same reason, the next co-efficient, wh 

formed of the products of any three of these ne 

quantities, must be negative ; and the next fbllowin 

ing made up of the products of any four of the sa 

^ative quantities, must be positive, and so on. 

And, from this reasoning, it plainly appears, that 
all the roots are fiositive^ the signs are filus and 
alternately. 

But if the roots be all negative, as jre=-— a, x^ 
3?=— c, xss— ^i then (x+a)x(x+iJ)X(jr+c)X(. 
aso, will express the equation to be produced; a 
the terms will plainly be positive. 

So that, when all the roots of an equation are neg 
M is plain that there can be no change in the signs ( 
terms 9f that equation. 

And, in general, there will be as many pc 
roots ia vxkj equation as there are changes ii 



flSC 



?/- 



121' 



NATURE OF EQUATIONS. 117 

signs of the -terms of that equation, from + to — , or 
rifl from— to +; and all the rest of the roots will be ne- 
L^l gative. 

From thb rule it follows, that, in quadratic equations, 
the two roots may be either both positive, or both nega* 
livC) or one negative and one positive. 

ThuS) in the equation, a ^ , + c^ = o, or (x — o)x 

Xj>— *), there are two changes of the signs, and therefore 
the roots are both positive. 

In the equation x^ J_ ^^ + ab^o^ or (j:+c)X(x+^), 

thei^ is ho change of the signs, and consequently they 
arc both negative. 

And, m the equation ^^ ■ ^ — 0^=0, or (x-— a) X 

C*+^)> otie of the roots will be affirmative and one ne- 
gative; for, as the first term is positive and the last ne- 
gative, there can be but one change in the signs, whe- 
\ Uicr the second term be -f or — . 

\ In cubic equations, the roots may be all positive, or all 
negative ; or two of them may be negative and one posi- 
tive, or one negative and two positive. 

Thus, in the equation (.r — a)y.{x — 6)x0r — r ) = 0, 
the signs will be alternately + and — ; and, as the 
number of changes is three, the roots must be all 
Ijositive. 

In the equation (or -f: a) X (J^ + *) X (j:* + c)=o, where 
there are no changes of the iigns, the roots must be all 
negative. 

In the equation :r2_(fl-_id4- c).a2 + (06 — ac — be) x 
4. abc = 0, or {^x^^a) X (r — A) x (^^ -f- t-he number of 
changes will be two, and conseqneiiiiy two of the roots 
Avill be positive and one negative. 

For if + ^ be greater than c, the second .term must 
be negative, its co-eflicient being — a, — 6, + ^ ; 



118 RESOLUTION OF EQUATIONS. 

and if a + A be less than c, the third term must be 
gative, its co-efficient + ab — ac — be being in that c 
negative. 

In the equation x^ -\' (a -{-b — c) x^ -f- («^ — «c — 
X'-^abc =ss Oj there can be only one change of the sig 
and therefore one of the roots is positive, and the otl 
tVvo negative. 

For if a + d be less than c, then the second terra 
negative, and the third must be negative also: and 
a + ^ be greater than c, the second term will be positi 
and there can be but one change in the other two ten 
whatever may be their signs. 

And, in the same manner, this reasoning may be 
tended to equations of higher dimensions, and theref 
the rule will be found to be true in all kinds of equati< 
whatever. 



PROBLEM L 

To increaae or diminish the roots of an equation by i 

given quantity** 

RULE. 

1. Take some new letter, and connect it with 
given quantity by the signs — or +, according as ii 
required to be increased or diminished. 

2 Substitute the powers of tiiis quantity in the eq 
tion, instead of the powers of the unknown letter, i 
there will arise a new equation, whose roots will be a 
mented or diminished as required. 



* When a cubic equation has two equal roots, it may alway 
reduced to a lower diaiension, and the solution, by that means, n: 
mare e»sy. 



RESOLUTION OF EQUATIONS. 119 



EXAMPLES. 

^ !• Let the quadratic equation x^ + 8jc + 15 = be 
given; it is required 10 increase its roots by 7. 

\ Suftfiose j:= y — 7, 
Then x^^y^ — 14^4.49 
8x= 4- %y — 56 
15» 4- 15 



y%^^Sy 4- 8s=o=srAff equation required*. 

2. Let x^ — /ix2 4- qx — r=io^ be the equation given ; 
it IB required to diminish the roots by the quantity e, 

Sufi/iose xsssy +€; 
Then x^ssyi+Sey^+Sehj+e^ "^ 
-^.-Z^Ss .^ fiy2 — 2/iey — /le^ I = Oj t^e new equation 
'\-qxss ^ qy^qc f required^, 

'^ r sss — r J 

3. Let x^+x^ — \0x 4- 8 srsO be given, and let its 
roots be increased by 4. 



* For, in the fonner equation jc34-8^+15s3=o, the roots are — »3, 
and —5, and in the equation y^^-fy-^-SssOt the roots are 2 and 4; 
therefore the difference is 7» as was required. 

t The last term of this transformed equation is the same as the 
^ven eauation, having e in the place of ^. 

And ^i<*^ this it appears that, if the last term of any equation is 
to be destroyed , the difficulty will be no less than that of solving 
the original equation itself. 



150 RESOLUTION OF EQUATIONS. 

Su/i/iose jc=sy~^4i; 

Then x3=y3 — l2z/2 + 48i/ — 64 

+ .r2= 4- y2 — 81/+15 

— I0jt7s= — 10I/+40 

+ 8 = +8 



iMdM^Midh*iMi^>^M*^*i 



5k« aat/5 — llt/2+30y=«sO, 
Or 2/^—1 ly+30=30, /A^ equation tequin 
In ivhich equation y is found aes 6 j anrf conaequ 



X s=:2. 



PROBLEM n. 

To take away the set and tetmfrdm any equation 



RULE. 



L Divide the co-efficient of the second term bj 
index of the highest power of the unknown quantity; 

3. Annex the quotient, with its sign changec 
. some new letter, and this, being substituted for its e 
in the given equation, will destroy the second tetn 
required. • 



* In thiB example, the -given equation is reduced to a quadi 
sftidin the prrB(.nt case, as well as in ail others, where the last 
vanishes^ the number assumed (•^) is one of the reots of the 
pos«d equation 

The- affirmative roots of an iiquatbn are thangie^ ' ' *-f<i: 
ones of the sami- \'alue, and the negative foots iiito lf& r . .ve< 
by only changing the signs of the terms altetnately, bbgumiiig 
the second. 



RESOLUTION OF EQUATIONS. 121 



« EXAMPLES. 

1. Let the quadratic equation x^ — 8x + 15 = be 
giTen; it is required to take away its second term. 

Skftfiote x=sy +4(y4-|); 
Then x»=ya+ 8y + 1 6 
— 8x= — 8y — 32 

+ ISas +15 



y* — \=^o=:equatton required*. 

2. Let the equation a:^— 9ji:2+26x — 34s=0 be given; 
it is required to exterminate its second term* 

%Me X =sy +3(y+|); 

Then x3=y3^9y2_|.27y+2r 

^ 9x«== — 9y2 — 54t/ — 81 

+ 26^: = +26^4-78 

—34 » —34 



y3_y«. XQrszo^si equation required. 



Thus the roots of the equation x4— x3— 19x3-j-49x— 30==o are 
-j-l, -(-2, +3, — 5 ; but, by changing only the second and fourth 
terms, the equation becomes x4-f.x3 — 19x3^49x — 30=0, and the 
toots are —1, —2, —3, +5. 

All the roots of an equation may also be made afBrmative or ne. 
gative, by increasing or diminishing each of them by some known 
quantity. 

* From this example it appears, that any quadratic equation may 
be solved without completing the square, by only taking away the 
second term ; for since ^*=1, or jrsssy/l^l, we shall have jcmey-f- 
4^14-4«b5, the root required. And the same may be shown of 
any otlier adfected quadratic equation whatever. 

L 



122 RESOLUTION OF EQUATIONS. 

3. Let x*+8x3 — 5j?2+10jc — 4=0 be given; to ex- 
terminate the second term*. 

Suppose X =y — 2(y— |); 

Then ar4=z/4_8i/3+24i/2— 32y+16 
+ 8x3= -|.8t/3— 48i/*+96y — 64 

— 5^2= — 5y2+20i/ — 20 
4-10x= 4-101/— 20 

— 4 = — 4 



^ 1/4 — ^292/2+941/ — 92 =^o = equation re* 

quired, 

4. Let x^ — /2x3+yx2 — rx4-«=0 be given ; to exter- 
minate the second term. 

P 
Suppose x^^y'\'-^\ 

4 

Then ^=y*+fiy^+ _LL + £| + ±- 

rp ^ ^ 

+ &• = 4- « 



• Since the sura of all the roots, in any equation, is equal to the 
coefficient of the seccnd term, it follows that, when the second 
term is wanting, the equation has both affirmative and negative roots, 
and that the sum of the affirmative roots is equal to the sum of the 
negative ones. 

Thus, in the cubic equation x^ — 7xssi6, .the three roots are -f-3, 
— .J2/ and — 1, where it is evident that 3ss2-|-l. 



RESOLUTION OF EQUATIONS. 123 



PROBLEM HI. 

Tojind whether some or all the roots of an equation be 
rational; and, ij'aoj what they are, 

RULE*. 

I. Find all the divisors of the last iermy and substitute 
them one by one for the unknown quantity. 

9. Then, if the positive and negative terms destroy 
each other, the divisor, so substituted, will be one of the 
roots of the equation. 

3. But if none of the divisors succeed, the roots are, 
for the general purt, either irrational or impossible* 

JVbte. When the divisors of the last term are too nu- 
merous, they may be diminished by changing the equa> 
lion into another, whose roots are augmented or decreased 
by a unit, or some other known quantity. 

EXAMPLES. 

1. Let ^3.^4 jp2 — 7j7 + 10 = be the equation pro- 
posed . 

Then the divisors o/*(10) the ia6t term will A^ + 1, — 1, 
+ 2, — 2, +5, — 5, + 10, — 10. 



* Since the last term, in any equation, is always equal to the pro- 
duct of all the roots in that equation, those roots must, therefore, 
necessarily be found in the number of its divisors. 

But this, it is evident, can hold only when the roots are commen- 
surate, or whole numbers. 



134 RESOLUTION OF EQUATIONS. 

/ 

And theae^ being aubatituted aucceaaively inateat 
will give 

I-^ 4— 7+10= 

— .!_ 4+7+10= 12 

8 — 16 — 14+10= — 12 

_8_ 16+14+10= 

125—100 — 35+10= 

There/ore + 1, — .*, and +5 are the three roota 

equation required. 

2. Let y^ — 4y3 — 8y + 32«0 be the equatio: 
posed. 

1« Change it into another^ the number of whoae d, 
ahall be Uaa; thua^ 
Su/ifioae y=x + 1 

Theny*=^x^+4x^+ 6x^+ 4x+ I 
— 4y3= — ix^ — 12x« — \2x — 4 
— By = — Sx — 8 

+32 = +32 



ac;^n,.^^x^'^l6x+2\ssiO=snew equati 

2. The diviaora of the laat term (21) o/* this neiv 
tion are 

1,— 1,+3,— 3, +7,— 7, +21, —21. 
And if these be subalitnted successively itisttad of 
a/iall have 

■ 1 — 6 — 16 + 21= 
1 — 6+16 + 21=32 

81 — 54—48 + 21= 
81—54+48 + 21=96 

tS^c, where none of the others sr, 
So that 1 and 3 aie the only rational rootay the oih, 
being im/ioaaible. 



* Note, The divisor of the last term of this new equation 
diiQinished in the same ffiscnner as before. 



RESOLUTION OF EQUATIONS. 125 

3. Let x^'^-Sax^''^a^X''-^\2d^s=siOf be the equation 
Bi*oposed. 

ffere the numeral divisors of the last -term ( 1 2a^) are 
I, _i, +2, —2, +3, ~3, +4, -^4, +0, —6, 

+ 13, — 12. 
-^^nd by substituting these successively instead ofx^ we sMll 
have 

1+ 3— 4 — 12=— 12 

—1+ 3+ 4—12=— 6 

8+12— 8 — 12= 

—8+12+ 8 — 12= 

27+27-12-12= , 30 

—27+27+12—12= 

Therefore the three roots are 2o, — 2flr, and — 3a. 



PROBLEM IV. 

To discover the roots ofequatioyis by Sir Iaaac 
Newton'* method of divisors. 

RULE. 

1. Instead of the unknown quantity, sul>8titutc sue- 
^ssfively three, or more, terms of the arithmetical pro- 

^t*ession 2, 1, 0, — I, — 2. 

2. Collect all the terms of the equation into one sunr, 
^\i«l place them, together with their divisors, in per- 
feendicuhir lines, ri^ht against the corresponding lerms^ 
^i* the pi'ogression 2, 1, 0, — 1, — 2, 

3. Seek amongst the divisors for an arithmetical pro- 
gression, whose terms correspond with the order of the 
terms 2, 1, 0, — 1, — 2, and whose common difference 
IS either a unit, or some divisor of the co-cfificieut of the 

. L 2 



i 



126 RESOLUTION OF EQUATIONS. 

fiighest power of the unknown quantity in the given 
equation. 

4V Divide that term of the progression, thus found, 
which stands against the term in the first progression) 
by the ratio or common difference* 

• » 

5, To the quotient last found, prefix the sign + or 
— r according as the progression is increasing or de- 
creasing, and this number being substituted for the un- 
known quantity, will be found to be one of the roots of 
the equation. 

JVbte. When there is more than one progression^ the 
roots must be taken out of each. 



1 



EXAMPLES. 

U Let x3 — x^ — 10x+6=5s0, be the equation pro- 
posed. 

Thetiy by aubatituting aucceaaively the terma ofthefiro* 
greaaion 3, 1,0, — 1, inatead ofxy the work will atand aa 
Jbllowa : 



\at firog, 

2 

1 



— 1 



reaulta. 
—10 
— 4 
+ 6 
+ 14 



diviaora, 
1.2.5. 10 
1.2.4 
1.2.3. ^ 
1.2.7. 14 



2rf. t^rog 
5 

4 
3 
2 



jlnd — 3, the term atanding againat o^ being aubaiitutedfor 
Xy givea^27 — 9 + 30+6=0; and therefore --^3 ia a 
r^ot of the equation. 

2. Let 2;r3-.— 5:c2+4j; — 10=0, be the equation pro- 
posed. 

Theuj by aubatituting aucceaaively the terma of the fir o^ 
greaaion^ 2, 1, 0, —1, — 2^inatead qfx^ the work will atand 
aafoUowa: 



RESOLUTION OF EQUATIONS. 127 



\8t fir og., results. 







— 6 

— 9 

— 10 
—21 
— 54 



divisors. 
1.2.3. 6 
1.3.9. 
1 .2.5. 10 
1.3.7.21 
1 .2.3. 6.9 



2d firog 
1 
3 
5 
7 
9 



Utftro. 


results. 


2 


70 


1 


144 





180 


•-r 


160 


—2 


90 



progressions. 



Here 5f the term standing against o, being divided by 2, 
the common difference^ gives 2^\ and this being substitut' 
fd for xj gives S\^ — 31^+10 — 10= o; and^ therefore^ 
^\iMa root of the equation, 

3. Ut j:^+x^ — 29x* — 9x-f 180r=o be the equation 
proposed*. 

Tkeuy by substituting as before^ the work will stand as 
foUovn: 

divisors. 
'1.2.5.7.10,14, isfc, 
1.2.3.4. 6. 8, ^c. 

1.2.3.4. 5. 6, ^c. 

1.2.4.5. 8.10, isfc. 
1.2.3.5. 6. 9y ISTc. 

^^ that here are four progressions^ and the numbers 3, 4, 
■*-3, and —5, being each substituted for x, make the whole 
^9^tQtion vanishy and are therefore the roots required. 

4. Given a.^ — Sa-^ — 46x — 72=0, to find the values of 
^'» by the method of divisors. j^jis +9^ — 2, and — 4. 

5. Given x* — 4>x^ — 19:c2+46x+ 120=0, to find the 
Values of Xy by the method of divisors. 

j^ns. +5, +4,-3, —2. 



1 

2 


2 
3 


5 
4 


7 
6 


3 


4 


3 


5 


4 


5 


2 


4' 
3 


5 


6 


I 



* Several other rules for discovering the roots of equations may 
^ found in Newton's amd Madaurin's Algebra. 



1Q» RESOLUTION OF EQUATIONS^ 



PROBLEM V. 

To find the foots of cubic equations^ according to the me* 

thud q/*CARDAN. 
RULE*. 

\. Take away the second term of the equation, By- 
problem second, and it will be reduced to this form: 

2. Substitute the values of a and by with their proper 
signs, in. the following expression, and it will give the 
root required. Thus : 

r^uired. 



* The rule, from whence this method is derived, is xss 

^i^+%/(i^+-5V«3) + ^i^^s/iii^-^-^y. and the investigju 
tibn of it is as follows : 

Let the equation, whose root is required, be x^^ax^^b. 

And assume J' >|-2=r:x, and Sj'Zs — a. 

Then, by substituting these values in the given equation, we'shall 
Ivave;^3_|-3;a24-3;'«2+«34.aX(;'+2)=;'34-23-f37«X(7+2)+fl 

And if, from the square of this last equation, there be taken 4 
times the cube of the equation yz^:^ — ^a, we shall have^fl 2^3^^ 

But the sum of this equation and {y^^z^ss^b) is 2y3--.4 i • 
(52-|-^yi3) and their difference isSz^sa^— ^(^«|. 4a3)j wlience 

y is found =g^^/>4.^( ;^i>a-f ^i^,and g=a 



RESOLUTION OF EQUATIONS. 129 

^ote*. When a Is negative, and -^a^ is greater 
than ^a, the solution, by this rule, cannot be generally 
^*tained.'. - / 



EXAMPLES. 

. i. Let y3+3y«+9y=13, be the equation proposed; 
It is required to find the value of y* 

!• In order to destroy the second term^ letyssx-'^l ; then 
y»aBBx3 — 3x2+3jr — 1 
3ya= +3j:«— 6:c+3 
9y = +9ar— 9 



x^+6x— 7=13 
or, j:3+6xa=20 



And from hence it appears, that y-^-z* or i ts equal x, is = 
^iA+^(i*»+ 1 a3) + ^i^— ^(i^-f 1 a3), which is the theo- 

Or, since « is ss — A it will be j>+a;s==)f— -2L«_v 

3;^ ^ 3y — 

'^^A-j-^(^^4- 1 tfS)— — "^^ , f Ae tffltme as the 

This method of solving cubic equations is usually ascribed to 
Oardan : but the invention is not his. The authors of it were Scipio 
^trreiu, and Nicholas Tttrtalea^ who discovered it about the same 
^ime, independently of each other, as is proved by M. de Montucloi 
Ui his JBUtoire des MathSmatiques. 

• This is called the irreducible case ; and, notwithstanding several 
of th^ most eminent mathematicians in Europe have attempted the 
Bolotion of it, no general rule has yet been discovered. 

The usual method is by a taWe of sines, or by throwing the ex- 
pression into an infinite series, and finding the sum of a certain 
number of termst according to the degree of exactness required. 



■\ 



130 RESOLUTION OF EQUATIONS. 

S* jFor a fiut 6, and for b %0, and wis shajii have 



^=^|6+^Q.2-f^V«^)--- 



4« 



2 



-r\^v -1- J ^10-1-^(1004-8) 



^104.10.3923-^|7=====3 «^20.3923_ 

2 2 

=2.732— ——-=2.732— .732 =2; /Aa/ m, 



^20.3923 2-^32 

ji:=:2, and consequently y=sl s=root requited, 

3. Given j:3 — 6x= — 9, to find the value of j:. 
Here a=s — 6, and b = — 9 ; and therefore we shall have 



i" 



^=n*+v?a^^+^a^'-:^^^^p^3) 



— 2 



— 2 

— -=-— 1— .2=— 3 ; fAaf w, ar=— 3= roo^ required. 



EXAMPLES FOR PRACTICE. 

1. Given a:^—- 6x^4- 10x=s8, to find x. Ans, x=4. 

2. Given y34-30y---n7, to determine y. An%^ ys=3. 

3. Given . 1/3— i.36y=91, to determine y. ./^n«. y=7. 

4. Given y^ — 3j/=18, to determine y. vf;?*. y=3. 

5. Given y34-24y=250, required y. ./^w*. y=5.05. 
d. Given y« — 3y4 — 2y*— 8=0, to find y. Ann. y=2. 



RESOLUTION OF EQUATIONS. 13 1 



PROBLEM VL 

Tojind the roota of biquadratic equationay according to the 

method o/Des Cartes. 

RULE*. 

1. Take away the second term of the equation by 
problem 2, and it will be reduced to the form x^-^-qx^-^ 

2. From the cubic equation y^+'2qy^+{q^^-^8)y — r^ 
=0 take the second term, and find the value of y by the 
last problem. 

3. Let e be assumed =x/y,/=j7+^e2 — and ^= 



2(? 



*^+4^*+^' 



4. Find the roots of the two quadratic equations x^-^ 
<rj-4/=0, and x^ — ex-f-.§-e=0, and they will be the four 
roots of the biquadratic required. 



• Irvceftigation of the rule. Let the given equation x^-|-9'^+^ 
-f^=sO be equal to the product of the two quadratic equations x^-f- 

Then, by equating the homologous terms, we shall havey-f ^ — 

r 
t^ssqt eg-^fssr, zndfg^^s; and therefore /=? 9+ ^ea—--, g=: 

k+i««+^. and s:=:fxg=:qi+iqe2+le4^^. 

Aq4 from this last equation we shall have e^^^qe*J^(jg2 — is)x 
d^^sT^-ss to a cubic equation, in which the value of e may be fdtfnd, 
as in the last problem. 



132 RESOLUTION OF EQUATIONS. 



EXAMPLES. 

1. Let z^— 4z3 — 8z+33sbO be the equation pr 
in which it is required to find the value of z. 

1. To take away the second term^ let x+ l=z; 

2:4=J74 + 4j73+ 6J7»+ A,X+ 1 

— 423 — — 4 j;S — 1 2x« — 1 2x — 4 

~.8z 8 — 8x — 8 

• +32 SB +32 



j?4.*_ 6j?2 — 16X+21 =0, or 

T/3 — 1 2ya — 48y — 256=xO,/or *Ae cm^/c e< 

2. To take away the second term from this egtui 
fi+4i=sy; then 

y3^fi^+i2/i^+4Sfi+ 64 
— 12z/«= _12/i»— 26/i— 192 
— 41y = —4>Sfi—\92 

— 256 = — 256 



/i3— .96/^=576 



Bi|t/(=ig+ic2-^) and g (=ri7+ie»+-^) arc als( 

and therefore the roots of the quadratic equations x2+ex 
and x2— «x+^=so, may be determined, and are the four 
the biquadratic equation required. ^ E. L 

Note. The co.eificient of x is put equal to e, in both t 
tions, because, when the second term is wanting, the su 
positive roots is always equal to the sum of the negative oi 
a contrary sign. 

THis rule has sometimes been ascribed to Dm Cartes^ a: 
times to Bomdelli, an Italian ; but the original inventor « 
Xouis Ferrari. 



RESOLUTION OF EQUATIONS. 133 

S. Tojind the value offi^ by Cardan's ruiefor cubic 
^mtiona. 



—32 



A 1 g 

^^^fii and therefore ^=16, or^yssi4y /= -— -\ 

. 16 , — (5 16 16 

f^7jandg=sS. 



^ Inthe method oiDes Cartes, above explained, all biquadratic equa- 
tions are supxxwed to be generated from the multiplication of two 
Jtiadratic ones : but, accoiding to the following way, which is taken 
ntmi ^mptof^s Algebra^ every such equation is conceived to arise by 
taking the difference of two complete squares. 

Here, the general equation x^-f.a)^4'^^~l~^^H~^=^ being pro- 
pOied, we are to assume (x8-f-ifl3f+ a)^ — (Bjc+c)2=rf(r*-^flx3^c* 
'\4 ; in which a, b, and c, represent imknown quantities to be 
detemuiied. 

Then x*-f ^a*+A, and bx+c being actually involved, will 
give x^+oxa+SAx* "1 

-f ia»x24-aAx4- A* S. =sx*-j-ax3+^xa 

— B«jc2— 2bcx— c« 3 
-^cx^d: from whence, by equating the homologous tenns, we shall 
aave j 

1. 2A4.|aS -.bS=:6, or 2A-f ja^— A=rB*; 

2. aA— 2bc . =sc, or aA— c s=:2bc; 

3. aS— cs ssflffOr aS^^/ =c3. 

Let now the first and last of these equations be multiplied together, 
and the prodnct win, evidently, be equal to \ of the square of the 
second; that is, 2A3lf (^a*— A)XAa— ^/a— (ifl«— ^)X</=(b2c2) 
Bi X^A*-— 2acA-|> cs. 

M 



134 RESOLUTION OF EQUATIONS. 

4, To Jind the roots of the two quadratic egua 
x^^ex-^-fsssOj and x*^— -ex+^=o. 

:r^+ex+/s=x^+^x+7=:0, 
x» — ex+g^x^ — 4x-|-3=0. 

In the Jirat of these x= — ^+\/ — 3, or — 2 — v 
And in the second J7=i3, and 1. 

. Therefore^ 3, 1, — 2 + ^ — 3, or — -2 — -v/ — 3, ai 
four roots of the equation x^ — 6x2— -l6:c+21=o. 

Jind if unity Be added to each of them^ we shall 
4, 2, — \+y/ — 3, and — 1 — y/ — 3, for the roo 
24— 4z3— 8z+32=o, the equation fir ofiosed ; the tw 
of which are im/iosaible, 

2. Given x^+2x^—7x^ — 8x4-12=0, to find 
values of x, Ans. x= 1, 2, — 3, and 



Whence, by denoting the given quantities ^ac — d, and^c^ 
(|a»— ^) by i and /, respectively, there will arise this cubic equ 
A* — ^AxS-j-iA— ^/ssso ; by means of which the value of a nr 
determined ; and therefore, from the preceding equations, I 
and c will also be known ; b being found from thence =v^( 

flA — c 

Ja2-^), and Css-ge" ' 

The several values of a, b, and c, being thus found, tha 
will also be readily obtained : for (x2-|-^i2x-|-a)2 — (bx-j-cJ2 
universally, in all circumstances of x, equal to x^-f-ox^-f-ox 
-^df it is evident, that, when the value of x is taken such th 
latter of these expressions becomes equal to nothing, the formei 
likewise be equal to nothing ; and consequently (x2-j-Jax-j-j 

(Bx-|-C)2. 

And therefore, by extracting the square root of both sides < 
equation, we shall have x»4-ia^-f-A=d:Bx±c ; or x=±i 

a±l (ifl±^B)2dtc— a],*=:±4b— ^a± (TV«^±iaB-f-iB2±c- 
which exhibits all the different roots of the given equation, ac 
ing to the variation of the sigp:is. 



RESOLUTION OF EQUATIONS. 135 

3. Given x* — 25x2+60x — 36=0, to find the values 
^^^' Ans. 1, 2, 3, and — 6-. 

4. Given y^ — Sy^+XAiy^+Ay — 8=0, to find the values 
ofy. 

jlna. y=3 + ^5, 3 — -v/5, 1+^/5, awe? 1 — ^/3. 

5. Given x^+ \2x — 17=0, to find the values of x, 
-f«». j:=ix/2±v^( — 3v^2 — J), and also — i\/2± 

^(3^2— j). 

6. Given x^-^^^x^-^Sx'^ — 4a:+ 1=0, to find the va- 
lues of x. ^ — 1±^/— 3 ^5±x/2l 

^ns, x= < and • , 

2 ' 2 

7. Given x* — \Ox^+SSx^ — 50a:+24=0, to find the 
Values of X, Ans, l, 2,3, and 4. 



This method will be found to have many advantages over that 
jiven above. In the first place, there is no necessity for being at 
the trouble of exterminating the secoM term of the equation, in or- 
dcr to prepare it for a solution ; secondly, the equation a3 — ^^a^-j- 
^A— ^/=:o, here brought out, is of a more simple form than that 
derived from the former method : and, thirdly, the value of a in 
this equation will always be commensurate and rational t not only 
when all the roots of the given equation are commensurate, but also 
when they are irrational^ and even impossible. 

Example. Let there be given x^-\-\2x — 17=0, to find the value 
ofx. 

Here, by comparing this with the general equation x^^ax^^bx 
-\-cx^d^o, we shall have a=(7, b=iO, c=sl2, and «^=ail7; and 
therefore kzs^iae — d^Vr, /=4c8-f-rfx(4«*— ^)=36, and a3— 
iAA2-fifA— i/=A34.17A— -18=0. 

And, from this equation, a will be found equal to 1; and there- 
fore B=(2A4.^a2-^)^=^2, c=^^=^^ =— 3v'2» and 

x=s±}^2±(iT3v/2— l)*=±is/2:f(±3v'2— i)^. 



136 RESOLUTION OF EQUATIONS. 

8. Given x«—^j73—58j?2 — lUx— .llasO, to find the 
values pf X. 

9. Given j?^— 3x*— 4:c— -3=0, to find the value of oc, 

. l±x/3 _l±v^_3 
Ana* x^ss. ^ana . 



PROBLEM VII. 

To find the roots of equations in general^ by the method 
of afifiroximation and converging aeries, 

RULfe*. 

1. Find, by trial, a nomber nearly equal to the root 
required. 



In some particular cases of this rule, the roots may be found by 
means of quadratics only. 

Several other methods of solving biquadratic equations have been 
invented by different authors ; one of the most ingenious of which 
is that given by M. Euler, in p. 664 of his EUmena d*Algebre. 

Equations of five or more dimensions may sometimes be reduced 
to those of an inferior degree ; but the process will be exceedingly 
tedious, as no general rule can be given for resolving them. 

• The rules hitherto given for finding the roots of equations are 
either very troublesome and laborious, or else confined to particular 
cases; but this method, by converging series, is universal, extending 
to all kinds of equations whatever; and, though not accurately true, 
gives the value sought to any assigned degree of exactness. 

The method of obtaining the roots of equations by approximation 
was first made use of by Vieta. 



RESOLUTION OF EQUATIONS. tST 

2. Call the number, thus found, r, and let z be put 
equal to the difference between r and the true root x, 

8. Instead of x, in the given equation, substitute its 
equal rdizy and there will arise a new equation, affected 
only with z and known quantities. 

4. Reject all those quantities in vhich there are two 
or more dimensions of z, and the value of z will be 
found by means of a simple equation. 

5. Add the value of z, thus found, to the value of r, 
and it will give the root required neaHy. 

6. If this root is not sufficiently near the truth, re- 
peat the operation, by substituting it instead of r, in the 
equation exhibiting the value of z, and it will give » 
second correction for the root required. 



EXAMPLES* 

1. Given x2 — Sx — 3J=:0, to find the value of j: bjr 
approximation. 

The root J found by trialy «> nearly equal to 8"; 

X<?r, therefore^ 8=r, and r'\-z^=^x ; then 

x2=r2+2rz+22 

_5 j7 =—5 r — Sz 
— 3l=— 31 



r2+2rz — Sr — 5z — 31=0/ 

3t4-5r— r» 31+40 — 64 7 ^ , 

«jr zs= — = — • = — =»6, and x^=^ 

2r—5 1 6—5 11 

8.6 nearly. 

Andy again, ifS.6 be substituted in the filacg o/r in the 

last eqttation, tve shall have 

51+5yw.r> 31+43—73.96 ^ .04 _ ^^^^^ 

^™ 2r — 5 "~ 17.2 — 5 12.2 * ^ 

and j:sb8.6+.0033s8.6032 nearly. 

M 2 



138 RESOLUTION OF EQUATIONS. 

Andy tf this value be again substituted /or r, it will give 
ZS3.0000077808, and x=b8.603277808 ; and so on to any 
degree of. exactness* 

2. Given x^-\-x^-\'X^^Oy to find the value of x by 
approximation. 

The rooty found by trialy is nearly equal to 4; 
Lety thereforey 4ssr, a«dr+a:=x, theny 
x»a=r3+3r«z+3r2a+23 

X sar -i-z 



r3+3r2z-fr2+2rz+r+2=a90 ; 
96 — r3 — rg — r 90 — 64—16 — 4_ 6 __ 

®''' ^'^ 3ra+2r+l 48+8+1 57 ^^' 

ancf :i7ssb4.1 nearly. 

Jndagainy ifA,A be substituted in the place ofry in the 

last equationy we shall have 

90— r3-^r« — r 90 — 68.921 — 16.81—4.1 

' sss SS.00283, 

3r»+2r+l 50.43+8-2+1 ' 

and :cssa4.1+.00283sa:4. 10283 nearly ; and so on to any 
degree o/ exactness required. 

3. Given ^2+20^:= 100, to find the value of x by 
approximation. Ans, xs=4.1421356» 

4. Given x3+10x2+5ar=*2600, to find the value of 
^x by approximation. Ans. 1 1 .00673. 

5. Given x^+2x^ — ^23j7 — 70=0, to jfind the value 
of X, Ans» XS5.1349.. 

6. Given x^ — \Sx^+^x — 50=0, to find the value 
ofx. ^ ^n«. xsss 1.028039. 

7. GiVen x^ — Sx^— 75x=10000, to find the value 
of JC. jSns, xas7lO;2615. 

a. Given x»+2x4+3j?«+4*»+5x«a54321, to find 
the value of x. Ana. j3aPB8.4i44. 



> 



RESOLUTION OF EQUATIONS. 1S9 



RULE II. 

1. Assume the general equation az+bz^+cz^+dz^^ 
&c, =ssfi; where z is the converging quantity, and a, b, c, 
dy &c. co-efficients whose values are known. 

2. Then will ^ be an approximation of the secoftd 

degree. 

b c 

3. And, if 9 be put == —, we shall have 

a 

(a + afi) X A r . r • , . , • 

for an approximation of the third de- 



gree. 



4. And, in like manner, if w be put = h 7- , 

a d* -r ac 

then will — ^ ^ . ^^ . — V ' V / 7— r« ^ an ap- 

aX(a2+(6+ow)x/i)+(w — «)x/i2 *^ 

preximation of the fourth degree, &c. 



EXA.MPLES. 

1. Given a:«+20x=slOO, to find the value of a:. 

The rooty found by trials is nearly equal to 4; 

Lety therrforey A+z^sXy andy byaubatitutionytheequa* 
tion will become 28z+2*=4. 

Whencey from the ruley a ss 28, d = 1, c::sOy i^c. and 

aft 112 28 

Therefore z«= --i-— .»---=»—-««:. 142 1 3 for the 
^ a^+b/i 788 197 ' 

frst approximation. 



}40 RESOl-UTION OF EQUATIONS, 

* 

b c 1 
j^ndj since 9 = — — . -— s= — , ^e shall have z := 
. fl 6 28' 

{a+sti)y.fi ^ (2 84-|)X4 ^ 28+ n^ ^ 

«^+(^+tt«)X^ 28X38 + (I + 1)X4 28*-f-4-|-2 

-- — = .14213564yor ^^e second approximation, 
1386 

2d flrf — he 1 

wf/zcf. m /z^tf manner % since «;= 1 = — , it 

' ' a^ b^ — ac 14 

afiy.{a+wfi) 



will be z =s ' — 7-^-7-,, , s .V . . s — « — 

aX(a«+(<&+aw)X/i)+(w — *)X/i2 

28X4X(28+f) _ 28x(28+f) _ 28X198 _ 

28x(r84+12)+^ "" 7x796+4 ""49X796+1 "" 

5544 

= .1421356236; or or = 4.1421356236,- /or me 

39005 ' •^ 

roo^ required^ extremely near, 

2. Given x2+4.r — 10=100, to find the value ot x. 

Ans. 8,677078. 

3. Given x^— 17a:*+54jc=S50, to find the value of x, 

jina, 14.95407. 

4. Given x^ — 2x — 5=0, to find the value of x. 

jins. 2.094551. 

5. Given 2y4— 16i/3^40i/8— 30y=— 1, to find the 
value of y. ^n^. 2/=:l,284724. 

6. Given x«+2x4+3a:3+4x«+5a:=54321, to find 
the value of x. Ans, x=8.4 14455. 



In the same manner the roots of other equations may be approxi- 
•mated; but, to avoid trouble in preparing the equation for a solution*, 
all such powers of the converging quantity z^ as would rise higher 
than the degree or order of the af^rozimaiion you intead ta work 
by. may be f;^r^ where negleded. ^ 



RESOLUTION OF EQUATIONS. 141 



I 



PROBLEM VIII. 
To extract the root of any pure fioioer in numbers^ 

* RULE*. -• 

1. Let mmn the number whose root is required ; r= 
nearest root which can be found by trial ; and na to the 

index. 

2. Then, by putting v* -, we shall have 

* a* r -4 1— J i— ^ =root, nearly ; or :p «= r+ 

rxC2v+n) ^ , 

:rr7z — rx :^\ ^ \ — r — rr rr extremely near, 

^X(2i;+a;i— l)+iX(w— l)x(2n— 1) ^ 

EXAMPLES. 

1. Given x^^2; or, which is the same thing, let 
the square root of 2 be found. 

Supfioee the root found by trial to be 1.4 ; then we shall 

L ^ , . 2X1.96 ^^ 

have m«s:2. r=l.4, 72=s=2, anc? v=s -— =98. 

' ' ' 2 — 1.96 



* One of the most convenient ndes for practice, which has yet 
been discovered, is the following : X.'^•\-l)m-^^(n^l)v : (n-f 1)n-|- 
(fH- l)rn : : r : the true root nearly. 

Where it may be observed that n^ given number ; rss nearest 
root, found by trial ; and n=s index, as before. 



142 RESOLUTION OF EQUATIONS. 

j^nd^ therefore. x=tr4 ; --s=1.4+ 

lli^i££l=,.4-H-i2L.= ,.4 + -i£^= 1.41421356 = 
98X594 ^70X198 ^13860 

root required nearly. 

jlnd if the second afifiroximation be uaed^ the root will 
be found =zlA\42\356236i which is true to the last filac^ 
of decimals* 

2. Given x^^BOO ; or let it be required to extract 
the cube root of 500. . 

Sufifiose the root^ found by trial, to beS; then we shall 

3 V 5 1'2 ' 

harve ^=500, rs=8, n=ssS, and v=s _.ssb— -128; 

Mdy therefore, x^r+ ^^^ . f^^t "^"^^l =8— .063= 
' *^ ' vx(6v + 4>n) — 2 

7.93 for the first afifiroximation. 

Or x=:r4. rx(2T;+w) ^^_ 

■*"(2i;+2/i— Oxt'+^XCw— 1)X(2« — 1) 

6072 
— —-=7.937005259936 ybr the second afifiroximation, 

which is true to the last place of decimals. 

3. Let it be required to find the cube root of 2. 

Ans, 1.259921. 

4. Required the cube root of 117. Ans* 4.89097. 

5. What is the sursoUd, or 5th rool, of 125000 ? 

Ans. 10.456389. 

6. It is required to find the 7th root of 100000. 

Ans. 5.1794746792. 

7. It is required to find the S65th root of 1.05. 

Ans* 1.00013366. 



RESOLUTION OF EQUATIONS. 143 

PROBLEM IX. 

To find the root of an exfionential equation* 

RULE*. 

1. Find, by trial, two numbers, as near the true rodt 
as possible, ancf substitute them in the given equation 
uistead of the unknown quantity, marking the errors 
i¥hich arise from each of them. 

2. Multiply the difference of the two numbers, found 
by trial, by the least error, and divide the pixxiuct by 
the difference of the errors, when they are alike, and by 
their sum when they are unlike. 

3. Add the quotient, last found, to the number be- 
longing to the least error, when that number is too little, 
and subtract it when too great, and the result will give 
the^true coot nearly. 

4. Take this root and the nearest of the former, and, 
by proceeding in like manner, a root will be had still 
nearer than before ; and so on, t^ any degree of exact- 
ness required. 

EXAMPLES. 

1. Given j;*=100, to find the value of x by approxi- 
mation. 

By the nature of logarithms xXlog, x=slog. 100=2. 

./#«</, since x is found by trial to be greater than 3, and 
less than 4, 

Letj therefore^ 3.5 and 3.6 be the two su/i/iosed values 
ofx. 



* The rule for solving exponential equations was invented by '^• 
yean Bernoulli, and published in the Leipsic Acts, 1697. 



144 RESOLUTION OF EQUATIONS. 

Then the log. xsslog. 3.5 =.5440680; and xx 
log. xsss 1.9042380 
2 



— .0957620= r«r error^ too Utile. 
And the log. x=slog. 3.6=.5563025; a?id 
X X log, x=s2.0026S90 
2 
, , 4 

.0026890BB2e/ error^ too great, 
\8t number 3.5 Itt error —095762 

2d number 3.6 2d error +.002689 



QAxadiff. .098451=911111. 

0. IX. 002689 ^^„^« 

=.00273=correc/io». 

.098451 

2d number 3.60000 

correction — .00273 



3.59727 =x=roo/ nearly. , 
Againy sufifipae x=3.597; then we shall have 
log. x=. 5559404, and 
xXlog,x=sl,9997\76, which, subtracted from 2, 
gives .0002824, the third error, too little. 
2d number 3.600 * 2d error +.0026890 

3flf number 3.597 Zd error — .0002824 



.003 = diff. .0029 7 1 4 =««»!, 

,003 X. 0002824 ^^^^^ 

=.000285, the correction, 

.0029714 ' 

3rf number 3.597000 

corrfC//ow +0.000285 



3.597285 =jr=roo; required nearly* 

2. Given a:*= 123456789, to determine the value of .i\ 

Ans, x=8.640O268. 



[ 145 ] 



OF INDETERMINATE OR UNLIMITED 

PROBLEMS. 



A problem k said to be indeterminate or unlimited 
ivhen the equations, expressing the conditions of a ques* 
tion, are less in number than the unknown quantities to 
be determined. 

And though such Idnd of problems are capably of in« 
numerable answers, yet the results, in whole numbers, are 
generally limited to some determinate number, and may 
be obtained as follows. 

PROBLEM I. 

Tojlnd the values of x and y, in the equation axssby+ c; 
where a, A, and c are given numbersy which admit of no 
common drvi%or. 

RULE*. 

1. Let vjh stand for a whole number, and reduce the 
equation to x«» =wA. 

hv-4-c dv'hf 

2. Make =-£-!j:, by throwing all whole num- 

a a 

tiers out of it, till d and /be each less than a. 



* This nile is foanded on the following obvious principles : 
That the sum, difference, or product of any two whole numbers 

18 a whole number. 

And that If a number measures the whole of any number, and a 

part of it, it wiU also measure the remaining part. 

N 



146 OF UNLIMITED PROBLEMS. 

^ dy 4- f 

3. Subtract -^ ^, or some multiple of it, from 

ay 2ay Say 

— , — , , or any other multiple of y, that comes 

a a a 

near the former, and the remainder will be a whole num- 
ber. 

4. Take this remainder, or any multiple of it, from 
some of the foregoing fractions, or from any whole num- 
ber, which is nearly equal to it, and the remainder, in 
this case, will also be a whole number. 

5. Proceed in the same manner with this last rtfmain^— 
der ; and so on till the co-efficient of y becomes equal tOHi 

1; or^^^=Wi^.=A 
a 

6. Then will y = afi — g ; where /i may be any whol^ 
number whatever ; and as the value of y is now known^ 
that of a: may also be found from the given 'wmi^iiip. 

■ -w- 

EXAMPLES. 

I. Given I9xa=14y — 11, to find x and y in whoK. « 
numbers. 

14y— U ^ ^ 19v 

Firaty a?= — =5wA.; and -.-^sfp^. 

19 1^ 

^. z r ^ 19v 14t/—ll 5i+ll _ 

Then^ by aubtractioriy — — — «a - ■ ■ sk w^* 

^ . 5y + H 201/ + 44 20?^ + 6 . ' ^ 

30^4. 6 
i(ndj by rejecting the 2, — — — • as vfh. 

1 y 

Therefore — ^ — ^ *" "l9" ^* 

Andy^\9fi-^^\ vfhere^if fi be taken sx\^ fir the leat^ 
affirmative value t^ y> we •haU have y tm \3^ and x ■■ 9^ 
the answer. 



OF UNLIMITED PROBLEMS. 147 

2. GWen 3x=8y— 16, to find the values of x and ij iu 
*hole numbers. 

^ 8y — 16 ^ . 2y — I , 2y — I 

' 3^3 3 

2u I 4t/ — 2 3u 

^nd -^ X 2 =5 -^ = wh. But — h also = w//. 

,3 3 3 

»*,. ^ 4v — 2 3?/ y — 2 

-Therefore ^ = — -— = w/r. e=/j. - 

*^^ence y =± 3/i + 2 ; awrf Ay taking /i =i \j we shall have 
y ^* 5} and a: == 8, Me awswer. 



^. Given 9j; + 1 Sy s 2000, to find all the possible v^c 
'^ ^8 of X and y in whole numbers. 

2000 — I3y ^^^ . 2 — 4y 

2 — 4y 
Or, ^y rejecting 222 -r-y, — r — s= wA. number. 

3..4V 4 — 8y -' 9y 

^«rf, i X 2 = • = tvh. But* — 19 fl«o=3wA. 

'9 9 9 

TtA X ^y I 4 — 8.7 ?/ + 4 
Therefore,. -^ H ^ = = w/i. = /i. 

^P hence y = 9/i— 4 ; ant/, 6y taking fi = 1, we «/2fl// /iavt* 
y asB 5, a/acf jr = 2l5. 
jindy by adding 9 continually to the lait -value of y, and 
-^'Ubtracting \3from that of x^ all the fioasible anaivera nviU 
^tand aafcllowa : 

^ 5 215.189.163.137.1 11.85.59 33- 
** i202.176.l50.l24, 98.72.46.20* 
i 5.23.41.59.77. 95.113.131 
y "^l 13.32.50.68.86.104.122.140'''*^ 

4. Given 24:p= 13y+ 16, to find x and y in whole 
numbers. .dns. or = 5, and y = 8. 



148 OF UNLIMITED PROBLEMS. 

5. Given 14j: as 5y -f ^9 (o find x and y in whole nur 
bers. Ana, x =s 3, and y = : 

6. Given 27x = 1600 — 16j/, to find x and y in whoj 
numbers. Ana, ?/ = 19, and j;=4i 

7. Given 87ar + 256y==15410, lo find the values of - 
and 2/y in whole positive numbers. 

Ana. X = 30, and y = 5< 

8. Given 5^: + 7y -f 11 z =s 224, to find all the possibl 
values of x, y, and z, in whole numbers. 

Ana. The number ofananuera ia 5 ' 

9. To determine whether it be possible to pay lOt 
in guineas and moidores only. 

A?ia. The queation ia imfioaaibd 

10. How many different ways is it possible to pa 
IQOl. in guineas and pistoles only ? a guinea being equ 
to 21s. and a pistole to 17s. Ana. 6 different wa^ 

U. Forty-one persons, men, women, and childr^ 
spent among them 40s. of which each man paid 4s. eac 
woman 3s. and each child 4d. how many were there < 
each ? Ans. 5 men, 3 nvomeny and 33 children 

12. I owe my friend a shillino^, and have nothing abou 
me but guineas, and he has nothing but louis d'ors; tin 
question is, how must I acquit myself of the debt? tht 
louis d'ors being valued at 1 7s. Ana. I must fi'ay him 

13 guineas, and he must give me 16 iouia d'ors. 

13. It is required to discharge a debt of 3511. with 
guineas and moidores only, so that there may be the 
kast number of pieces of each sore; and to find what 
the whole will amount to, when paid every way it will 
possibly admit of. Ana. The number of ways is 37, 

and the whole amount 12957. 

•14. A vintner has wine at 2s., Is. lOd., and Is. 6d. 
fier gallon : how much of each sort must he take, so as 



OF UNLIMITED PROBLEMS. 149 

to make a mixture of 30 gallons, to be sold at Is. 8d. fier 
" gllloii? Ane. 16, 2, 12 ; 17, 4, 9 ; 18, 6, 6 ; or 19, 

8> 3, o/* each 9Qrt. 

.^5. To determine how many ways it is possible to pay 

lOOOI. without using any other coins than crowns, gui- 

•D^as, and moidores. Ana, 70734 different ways. 

PROBLEM II. 

. ^ojind sue A a whole number x, as being divided by t/ie 
J^^ numbers a, b^ c, ^c. shall leave the given remain^ 

• RULE* 

w f* Subtract ~ each of the remainders from a', and 
''^'Ue the difference by a, and there will result 

""'!; — y, M ' , , fjfc, = whole numbers. 

** . a a 

.^' Call the value of :r, in the first fraction, /i, and sub- 
f. ^^Ute this quantity in the place of x, in the second frac- 

. ^. Find the least value of /?, in the second fraction, by 
^^^ last problenii and call it r. 

4. Let the value of x be found in terms of r, and 
^.^bsiituttt this quantity in the place of jc in the third 
^^Qclion. 

5. Find the least value of r in the third fraction, and 
^ull it J? ; and the value of x in terms of *, being substi- 
'^Uted for x in the fourth fraction, and so on, will give the 
^thofc number required. 

EXAMPLES. 

1 . To find the least whole number, which, being di- 

N2 



150 OF UNLIMITED PROBLEMS. 

Yided by 17, shall leave a remainder of 7; but, beiog^ 
vided by 26, the remainder shail be 13. 

Let X ss number required. 

X — 7 X— 13 

Then , and = tvhole numbers, 

17 36 

X 7 

Andy by fiutting = ^, we shall have x = \7fi+ ^t 

Which value of x being substituted in the ^d Jracti^^i 

\7fi — 6 , » 26A . , 

gtves = ivh. But -—- is also =5 wA. 

* 20 26 

AndK therefore^ — — — = — ^z^oh ^ 

'z^ - 26 26 

Or ' X 3 = = /?-f ^-L — = wA. number-., 

26 2o 26 

Jlndj by rejecting /i, we have :» wh, =» r. 

Hence /izss^Sr — 18, and by taking rssl^we shall b> avc 

And<i consequently^ jr=»17x8-f7= 143, the number 
required. 

2. To find a number, which being divided by II, 19, 
and 29} the remainders shall be 3, 5, 10. 

L^t X = umber required, 

_, a: — 3 X — 5 . X — 10 , . 

Then — :— ? — r^— ' a"" — rr — = wAo/e numbers, 

X— 8 
vf«ff, by fiutting = /:, we shall have arss 1 1/} -f 3; 

Which value of x being substituted in the 2d fractiWy 
l^/_2 , n// — ^ 22/i — 4 




OF UNLIMITED PROBLEMS. 151 



=s vfh, Andy by rejecting /?, we shall have 

^, 3/i— 4 ^ 18/z— 24 18/r— 5 
' 19 19 19 

^ ^ wA. or by rejecting the 1, = wh, number, 

^^ —— Wj likewise* = wA. and. therefore^ — L — 
19 ' ' 19 

*8*~5 fi + S , , , 

— ■*: = — — — = wh, number, which let be fiut = r. 

19 19 ' -* 

2%«i,/i=»19r — 5, flnflf:c = (l9r — 5)Xll + 3 = 209r 

^52. 

•^7Z(/, dy substituting this value of x in the Zd fraction^ 

209 r — 62 ^ 6r — 4 
^'e shall have -^ = 7r — 2 H 5^= '^^^^ which, 

6r — 4 
'2^ neglecting 7r — 2, gives — -- — = wh, number: 

6r — 4 SOr — 20 r — 20 

^"^1 — :r:r~ X 5 = :» r -( — = wh. or^ 

• 29 29 29 

. . r — 20 

^y rejecting r, = wh, whidh let be fiut = s. 

Then r = 29* + 20; and, by /netting « = 0, we shall 
^iave r =s= 20. 

Andf consequently^ jr=:209x20 — 52=4128 ; 
Therefore i 4128 =s7iumber required, 

3. To find the least whole number, which, being di- 
vided by 19, shall leave a remainder of 7; but, being di- 
vided by 28, Che remainder shall be 13. Ans. 349. 

4. To find a nurnber, which, being divided by 3, 5, 7, 
and 2, will leave the remainders 2, 4, 6, and 0, respec- 
tively. Ans, 104. 



132 DIOPHANTINE PROBLEMS. 

5. To find the least whole number, which, being dl - 
vided by 16, 17, 18, 19, and 20, shall leave 6, 7, «, 9, KimJ 
10 remainders. jlna. 333550 

6. To find the least whole number, which, beingu 
divided by the nine digits respectively, shall leave n«z: 
remainders. jfna, 252(^. 

-7. To find the least whole number, which, being di- 
vided by 2, 3, 5, 7, and 1 1, shall leave 1, 2, 3, 4, and 5 fi^i- 
remainders. jin», 1623. 

8. To find in what year of Christ the cycle of the sun 
was 8, the cycle of the moon 10, and the cycle of indic- 
tion 10. Ms, In the year 1567, 



DIOPHANTINE PROBLEMS. 



* Diofihantine firoblems are those which relate to the 
finding of square and cube numbers. Sec. and are such 
as are generally capable of a great variety of answers. 
They are so called from their inventor, Diophantusy of 
Alexandria^ in Egyfit^ who flourished in or about the 
%hi»'d century, and is the first writer on algebra wc iiictt 
witli amongst the ancients. 

These questions are so exceedingly curious and ab- 
struse, that nothing less than the most refined algebra, 



, * That Diopha7itus was not the inventor of algebra, as has been 
generally imagined, is obvious; for his method of applying it is 
such as could only have been used in a very advanced state of the 
science. 

Hp no where speaks x)f the fundamental rules and principles, as 
an inventor certainly would have done, but treats of^ it as an art 
already sufficiently known ; and seems to intend, not so much to 
teach it, as to cultivate and improve it, by solving such questions 
as, befoxe his time, had been thought-too difficult to be surmounted: 



DIOPHANTINE PROBLEMS. 153 

applied with the utmost skill and judgment, can surmount 
^be difficulties which attend them. And, in this way, no 
'^an has ever extended the limits of the analytic art fur- 
^«er than Diofihantua^ or discovei^d greater penetration or 
judgment in the application of it. 

When we consider his work with attention, we are at a 
'OSS which to admire most, his wonderful sagacity and pecu- 
liar artifice) in forming such positions as the nature of the 
pit)blems requii*ed, or the more than ordinary subtility of 
^is reasoning upon them. 

Every particular question puts us upon a new way of 
thinking, and furnishes a fresh vein of analytical treasure, 
^bich cannot but be very instructive to the mind, in con- 
ducting it through almost all difficulties of this kind) when- 
ever they occur. 

The following method of resolving these questions will 
^ found of considerable service ; but no general rule can 
be given, that will suit all cases ; and therefore the Solo- 
mon must often be left to the sagacity and skill of the 
ieamer. 

It is probable, therefore, that algebra was known in the world 
long before the time of Diophmitus ; but that the works of preced- 
ing writers have been destroyed by the ravages of time, or the de- 
predations of ignorant barbarians. 

His arithmetics, out of which these problems were mostly collect- 
ed, consisted originally of thirteen books ; but the first six only are 
now extant The best ecUtion is said to be that published at Pdrit, 
by Monsieur Backet, in the year 1621 In this work the subject is 
so skilfully handled, that the modems, notwithstanding their other 
improvements, have been able to do little more than explain and il- 
lustrate his method. 

Those who have succeeded best, in this respect, are Vieta, Sraimc^ 
hr, Kertey, De Bilfy, Ozatmm, Prealet, Samderton, Fermat, and 
EaUer, The iMt o£ whom, in pBrticuUff has amplified and il- 
lustrated the Diophantine algebra in as dear and satisfactory a 
manner as the subject seems to admit. 



154 DIOPHANTINE PROBLEMS. 



.■■■>■ 



RULE. 



1. For the root of the square or cube required, put 
or more letters such, that when they are involved, ei 
the given number, or the highest power of the unkn 
quantity, may vanish from the equation ; and then, if 
unknown quantity be but of one dimension, the prob 
will be solved by reducing the equation. 

2. But if the unknown quantity be still a squ 
or a higher power, some other new letters must b( 
sumed to denote the root ; with which proceed as bef 
and so on till the unknown quantity is but of one dir 
sion ; and from this all the rest will be determined. 



EXAMPLES. 



1. 'To divide a given square number (100) 
two such parts, that each of them may be a sq 
number. 

Let x^ (=s □) Ac one of the fiarts^ and then 100- 
will be the other /lart, ivhich ia also to be a square nun 



Mf X — 10 had been made the side of the second squa 
this question, instead of 2x — 10, the equation would have 
x2.— 20:^+100=100 — x2 ; in which case, x, the^side of th« 
squaretNeould have been found aelO, and x— '10, or the side c 
second square =sO; and for this reason the substitution, x 
was avoided; but 3x — 10, 4xr«-l0, or any other quantity < 
same kind, would have succeeded eqiutUy as well as the foi 
though, in some cases, the results would have been less simplt 



DIOPHANTINE PROBLEMS. 155 

•^fsutne the side of this second square = Sx — 10. 
'^f^vnil 100 — j:2=(2j7 — 10)2=4ar« — 40:c+100; 

jindj by reduction^ j:=8, and 2x — 10 = 6, 
Therefore 64 and 36 are the fiarts required. 



THE SAME GENERALLY. 

I'tt a'ss given square number, x* (= D) = one qf ita 
P^ts^ and fl « j f ^rg ^Ae ofAer, which is also to be a square 
^*mier. 

Msume the side of this second square =irx-^aj 
ihen will fl« — x^ss^rx — a)2= r^x* — ^arx +a*; 

^ndy by reduction, x^t . owd ro? — as >, . — . a 

8flr* ar'-f-a ar^ — a 
^ : v . . — — r-: — =s —-- — = /o a square number. 

'^er^^ef — — J2a7irf|l— II-J2 ar^ the parts required; 
^htre a and r may be any numbers, taken at fileasure, 

3. * To divide a given number ( 1 3) consisting of two 
^own square numbers (9 and 4) into two other square 
'Umbers. 



If s and r be any two uneqaal nnmben, of which « is the greater; 
Den w3l Trs, «3— rs, and i^-^-r* be the perpendicular, base, and 
ypothemise of a right-angled triangle. 

And from this canon two square numbers may be found, whoee 
urn or difieicnce shall be square numbers ; for (3r«)s-)-(«s— rs)3iBx 
ft^r*)*, and (r»+#a)2— (2r*)s»(#«— r»)«, or (#>+rS)«— 
V .-rS)SaB(2r«)3 ; and this when t and r are any numbers what. 



* l%it question is considered by Disphantut as a very important 
net bdng made the foundation of most of bis other probtemi.—Jn 



1 56 DIOPHANTINE PROBLEMS. 

For the side qfthejirat aguare sought jfiut rx- 
for the side of the secondy sx — 2 ; r being the greater t 
her J and s the less. 

Then wHl{rx — ^Y+{sx — 2)aa«(r2a72 — ^rx +S 

(«2a72 — AiSX -f 4)=(r2+«2) j;2 — (6r + Aia)x +13=1 

(r2 -f «2)x2=(6r -f 4*)x. 

6r-f 
And from this equation^ by reductionj xisfoimd:=: — — 

6r2-(-4r» 3r2+4r* — 3«2 

Whence rx—3=z—P—- 3= ^^ , , = 

r2-f«2 r2+tf2 

of the first square sought. 

' r2-4-62 r2+«2 

of the second* 

So that if r be taken = 2, and « = 1, nve shall I 

3r2-f4r« — 3«2 17 era — 2r2+2*2 6 

1 =5 — , and ■ 3= --for the s 

r24-«2 5 » r2+*2 5 

of the squaresj in numbers^ as was required, 

Ifa^-^-b^ be put equal to the number to be divided^ 
ge?ieral solution may be given in exactly the same iham 



the solution of it given above, the values of r and s may be ta 
equal to any numbers whatever, provided the proportion of tl 
numbers be not the same as that of 3(a) to 2(A), or 3+2(a+^ 
3—2 (a — b). And the reason of this restriction is, that if r a 
were so taken, the sides of the squares sought would come cut 
same as the sides of the knDwn squares which compose the g 
number, and therefore the ojieration would be useless. « 

The excellent old Kertey^ after amplifying and illustrating 
problem in a variety of ways, concludes his chapter thus : «< F 
further account of this ra>^ speculation, see Jndersoau*, Theorei 
of Vieta'a mysterious doctrine of angular sections ; and likewise 
rigmiius, at the latter end of the first tome of his Curws Mathe 
fieus:* 



DIOPHANTINE PROBLEMS, isT 

% 
\ 

^. To find two square numbers, whose difference shaU 
^ equal to any given number (</). 

X'ft d be resolved into any two unequal fae tore a andbj. 
^ ^ting the greater and b the leaa. 

•^lao fiut xfor the aide of the lesa square sought j and 
^^bssiside of the greater. 

Then (x+b)^^^x*ssx^ +Ubx+ 6* — x2:ss2bx+b^x^d 

And if this be divided by bj we shall have 2x+bss:a. 

a — b 
Whence^ x = — 3 — asside cfthe less square sought^ 

a--^b o 4- 6 

and j?+d=s— +d3M— ^— aa5«rfe of the greater. 

So that by putting <fas60, anda+b^2x30, we shaU 

30-^3 304-3 

Jiave =14, and — ^=16, in- (i4)2= 196, and 

3 « 

(1 6)*=s356ybr the squares in numbers i and so for any dif- 
ference or factors whatever, 

4. To find two numbers suph, that, if either of them 
be added to the square of the other, the sum shall be a 
square number. 

Let the numbers sought Be x and y. 
Then x*+y*a q, and i/»+xss q, 

Andj ifr^-'^x be assumed for the side of the first square 
jp»+y, we shaU have jc«+y""''^^2rx+jc*, or y«r^— 
.3rx. 

Whence 2rx9ar*^^ ©r x^» ^ . 
Againj ify+9 be assumedfar the mde ffthe second square j 



WfihmUhaPitei/»+ Z7(y*+*)"(y+*)*«»y*+2ty+f«. 

O 



158 DIOPHANtlNE PROBLEMS. 

r*i— 2rs2 r^ — y 

r2 — 2r«2 , 2r294.«2 

So that and ■ are the numbers 

4r*4-J 4r*-|-l 

quired; where r and « may be taken at fileasureyfirovidei 

be greater than 2«2, 

5. To find two numbers, whose sum aiid diffeten 
shall be both square numbers. 

Let a: and x^ — x be the two numbers sought* 

Theny since their sum is evidently a square nufUbety (w^ic 
of the conditions of the question will be answered. 

There remains^ therefore^ only their difference x* — ^a:^ 
to be made a square. 

jitidy if for the side of this square there be put j?-i-r, we 
^hallhavex^ — 2rx+r^s=x2'^2xy or 2rx — 2jc=r*. 

tVhenee* J?=r: and x^ — a:= (- j i— . -— -, 

* 2r — 2 ^2r — 2^ 2r— 2 

r2 r2 v2 r2 

5o that anrf f ) ~ . are the numbers 

2r — 2 \2r — 2^ 2r — 2 

required ; where r may be taken at fileasureyfirovided- it be 

greater than 1. 

6. To find three numbers such, that not only the 
sum of ail three of them, but also the sum of eTery two) 
shall be a square number. 

Let 4!XyX^-^4tc^ and 2j:+ 1 be the three numbers sought* 
Then (4x)+(x«— 4ir)«ap«, <^3--4x)+(2x+ l)issa:a 

— 2j:+ 1, and (4j?+jr2— 44^+2^?+ l)=a:3+2x+ 1, are . 

evidently squares, 

Andy therefore^ three f^fthe conditigniynehtioned in the 
question aire accomplished. 



DIOPfJANTlNE PROBLEJyiS. 15? 

^^hence i/ remains only to make the quantity (4j?) + 

*^+ 1), or 6a: + 1 ass /o a square, 

"^er, therefore^ 6x4- 1 =a2; and we shall have x = 




•^«</, consequently^ , j — : — j* — 



4a2— 4 /a» — 1\ 4a2— 4 
??*•— 2 ." 2-2—2 a4 — 26/^2 4- 25 024.3 

^- +*' ^^— r-' — ^36 — ' '^"^ --r"*""" 

'*« numbers required: where a may be taken at /lieaauref 
^fovided it be greater tfian 5. 

J^, To find three square numbers such that the sum 
^ ^very two of them shall be a square number*. 

Let a:*, y*i andz^ be the numbers sought ; 
7»fnx8+22=a, T/»+22-= D, and x^'{-y^is:iU. 

^ X «2_ 1 y r2_ 1 

^ndy hy fiutting — ^: -: , and — = — - — ^ we shall 

7m jis z ^r 

2* '^ 4«2 ' 22 ^ 4r2 

^vMch are both evidently squares; and there/ore it remains 

x^ y* 
o«/y to make 1- — =s square number. 



* This question is capable of a great variety of answers ; but the 
least roots, which have yet been ^und, in whole numbers, are 44, 
lir, and 340. See El4men» d^Algebre, par M. Euler, tome 11, page 
327; which is a work particularly calculated for the use of those 
who wish to obtain a knowledge of algebra without the assistance 
of a master. 



160 DIOPHANTINE PROBLEMS. 

■i -^i^s " \ ^ • ssiaqtiare JVd. 

X (r+ l)*x{' — 1)®=3 /o a square number* 

And^ by making r — 1 =«+ i> ^^ raB«4-2> w? «Aa// Aor 
(« +2)2 X(« + 1)«X(«— 1)*+«2X(»+3)2X(«+ 1)»= fo ■ 
square number. 

Or («+2)»X(* — l)»+«*X(«+3)»=- 2*4+8»3+6fi2--i» 
-f 4 as ^0. a square number* 

J^ovf^ let the root qf this square be <M*Mmtfrfa=|«*— •-! — 

Then 2s<+ Ss^+6s^ — 4/»+4«(J*a— «+2)«a=ff«^ 
f.3^5«*+«» — **-f4; or ^44.843„2j^4 — 1,3. or 2«-H 

^^tf«« «r«^ — 34, ancTras — 22. 

^>— I 575 ^ y r2_l 483 

And X = ■■ =s p > , cncr -^ = ass / 

2, 48/ z 2r 44 

^ 575 z ^ 483z 

48 ^ ^ 44 

/w order ^ thereforey to have the answer in whole nuni' 
bers, let z == 528, and we shatt have x = 6325, and y = 
5796, or 528, 5796, and 6325 Jbr the roots of the squared 
required. 

8. To find a number, x, such) that x + I and x — l 
shall be both square numbera. Ans, xss^ 

9. To find a aumber x such, that jr+ 128 and :r+ 192 
shall be both squares. Ans, x^ss97* 

10. To find a number x suchi that x^ + x and x^ — x 
may be both squares. Ans. f| 



DIOPHANTINE PROBLEMS. 161 

11* To find two numbers, a: and y^ such that x + y^ 
**+yi and y^ + x may be all squares. 

jiiis, x= J, and i/=^V' 

13. To find three square numbers in arithmetical pro- 

S'Cision. jlns, 1, 25, and 49. 

13. To find three square numbers in harmonica! pro- 
i^rtion. .4n8. 1225, 49, and 25. 

, U. To find three numbers in arithmetical progres- 
sion, such that the sum of every two of them may be a 
^Uare number. ji?is. 120-J-, 840^, and 1560^-. 

15. To find three numbers, such that, if to the square 
^^ each of them the sum of the other two be added, the 
ihree sums sliall be all squares. ^na, J, ^, and I, 

16^ To find two numbers in proportion as 8 is to 15, 
^i)d such that the sum of their squares shall make a 
^uare number. jins, 576 mid 1080. 

17. To find four numbers such that, if a square num- 
»*er (100) be added to the product of .every two of them, 
^he sums' shall be all squares. 

^na, 12, 32, 88, and 168. 

r 

18. To find two numbers such that their difference 
may be equal to the difference of their squares, and that 
Ihe sum of their squares shall be a square number. 

^na. ^ and ^. 

19. To find three numbers in geometrical proportion, 
such that every one of themi being increased by a given 
number (1^)) shall make square numbers. 

.4n&. 81, -J, flw^ifl^. 

20. To find two numbers such that, if their product 
l)e added to the sum of their squares,- it shall make a 
square number. ^na. S and 3, 8 and 7, IG and 5, ^c, 

21. To divide a given number (10) into four such 
parts, that the sum of every ilirec of them may be a 
square numl>er. .^»'f. Ij 6, ^^^ and 5fJ 

2 



162 DioPHANtlNE PROBLEMS. 

f2. *To find three square numbersi such that their 
sum, being severally added to their three sides, shall make 
.square nuipbers. 

•^'**- iHjh mt^ «icf f}f|Ji==r©o;* required. 

23. To iirid two nunit>ers, such tiiat their sum, being 
increased and lessenedf either by their difftrencei or the 
di{rei*ence of their squares, the sums and remainders shall 
be all squares. jina. ^ and -^ 

24i, To find two numbers' such, that not only each 
number^ but also their sums and their difference, being 
increased by unityt shall be all square numbers. 

^n9, 3024, and 5624 

25. To find three numt>ers such, that whether theii 
sum be added ixh or subtracted fromi the square of eacli 
particular number, the numbers thence arising^ shall be 
all squares. Anit, y^*, '^*, and y/. 

26. To find three square numbers, such that the sum 
of their squares shall also be a square number. 

J[n9. 9, 16, and ^. 

27. To find three square numbers, such that the dif- 
ference of every two of them shall be a square number. 

Ana. 485809, 34225, and 23409. 

28. To divide any ^en cube number (8) into three 
other cube numbers. Ana, ^^-^^ and \, 

29. Two cube numbers (8 and I) being given, to find 
two other cube numbers, whose difference shall be equal 
to the sum of the given cubes. Ana. ^^ and ^^. 

30. To divide a given number (28) composed of two 
cube numbers (2T and I) into two other cube numbers. 

j/n,. tf^JJH ««' mum 'A* root,. 

31. To find three cube fiundbers, such that, if from 
every one of them a given number (l:}'be subtracted, the 
sum of the remaindets shall be » square. 

* The answers to many of diese ^lestioiis caxmot be given in 
whole nuinben. 



SUMMATION OF SERIES. 163 

!1. To find tiiree numbers such that, if they lie seve- 
lUf added lo ihe cube of their sura, the lliree suras 

'hence ai'ising shall be all cubes. 

^d. lo lind three numbers m ai'itbuietica] proportion, 
•icli thai the sum of iheir cubes shall be a cube. 

Jint. 3, 4, S, or 149, 236, 363, We. 
"4. To lini] three cube numbers, such that thdi* sum 
'i^l be a cube number, 

^'is. 3^ 4\anrfi3, or 2P, 19^, IB 3, ere 
^i. To tind two numbers such that theii' sum shall be 
'I'lal to the sum of their cubes. ^ns. tand&. 



SUMMATION AND INTERPOLATION 



INFINITE SERIES. 

Th* docirine of injinilc ecriet is a subject ivbicli has 
B^g;cd the attentioti of the greatest mathematicians in 
dl ages, and is, perhaps, one of Ihe most abstruse and 
difiicult branches of abstract mathematics. 

To find the sum of a series, the number of ivhosc 
lerms is inexhaustible, or infinite, has been considered 
iiy some as a paradox, or a thing impossible to be doae. 
But tliis difrictilty will be easily removed, by considering 
lilt every finite magnitude whatever is divisible in rij^- 
:iim, or consists of an infinite number of parts, whose 
L^'gregate, or sum, is cijual to the quantity first pro- 
posed- 

A number actually infinite is, indeed, a plain con- 
Crmltction lo all our ideas; for any number which wc 
can possibly conceive, or of which we have any notion, 
most always be determinait: and finite i so that u 
greater may be ntill assigned) and a greDter alter tliis; 




1«4 SUMMATION OF SERIES. 

and so on, without a possibility of ever coming 
of the increase or addition. 

This inexhaustibility, in the nature of nui 
therefore, all that we can distinctly compreheni 
infinity; for though we can easily conceive th 
quantity may become greater and greater wit 
yet we are not from thence enabled to form ai 
of the ultimatum^ or last magnitude, which is 
of further augmentation. 

We cannot, therefore, apply to an infinite f 
cx>mmon notion of a sum, or a collection of sei 
ticular numbers, which are joined and added 
one after another; for this supposes that th( 
culars are all known and determined. But 
series generally observes some regular law, and 
ally approaches towards a term or limit, we c 
conceive it lo be a whole, of its own kind, an 
must have a certain real value, whether that val 
terminable or not. 

Thus, in many series, a number is assignable 
which no number of its terms can ever reach, c 
be ever equal to it : but yet may approach to it 
manner as to want less than any given difiereni 
this we may call the value or sum of the series 
being a number found by the common method 
tion, but such a limitation of the value of the seri 
in all its infinite capacity, that if it were possib 
all the terms together, one after another, the su 
be equal to that number. 

Ag^in, in other series, the value has no limitat 
this may be expressed by saying, that the sum o 
riesis infinitely great; or, which is the same th 
it has no determinate or assignable value, but 
carried on to such a length, that its sum shall ex 
given number- whatever* 



SUMMATION OF SERIES. 165 

According to the common rule for- summing up a 
&ute progression of a geometric decreasing series> 
^here r is the ratio, / the greatest term, and a the least, 
^ sum is (rl — a) -5- (r^i) : and if we suppose c, the 
^ extreme, to be actually decreased to 0, then the sum 
of the whole series will be r/-s-(r — 1): for it is demon- 
s^le, that tlie sum of no assignable number of terms 
^'f the series can ever be equal to that quotient ; apd yet 
^ number less than it will ever be equal to the value of 
the series. 

Whatever consequences, therefore, follow from the 
Opposition of r/-^(r^ 1) being the true and adequate 
^ae of jLhe series, taken in ail its infinite capacity, as if 
^ the parts were actually determined, and added toge- 
^r, they can never be the occasion of any assignable 
pitor, in any operation or demonstration where it is used 
i<^ that sense; because, if you say that it exceeds that 
^alue, it is demonstrable that this excess must be less 
than any assignable difference, which is, in effect, no 
difference at all ; whence the supposed error cannot exist, 
)Dd consequently rl-i-^r — 1) may be looked upon as 
repressing the adequate and just value of the series, con- 
inued to infinity. 

But we are further satisfied of the reasonableness of 
bis doctrine by finding, in fact, that a finite quantity 
ictually converts into an infinite series, as appears in the 
ase of circulating decimal!^. Thus^ -f, turned into a do 
limal, is = .6666, &c. = ^ + ^ + .^^ + r^j^y &c. 
ontinued ad injiuitum, fiui this is plainly a geometric 
eries, beginning from -^, in the continued ratio of 10 to 
, and the sum of all its terms, continued to infinity, will 
vidently be equal to |, or the number from whence it 
iras originally derived. 

And the same may be shown of many other series, and 
if all circulating decimals in general. 



166 SUMMATION OF SERIES. 



PROBLEM I. 

v^/iy series being given^ to find the several order 

differences, 

RULE. 

1. Take the first term from the second, the sec 
from the third, the third fi-om the fourth, £cc. and 
remainders will form a new series, called the first oi 
ofi differences, 

2. Take the first term of this last series from the 
cond, the second from the third, the third from the fou 
&c. and the remainders will form another -new sei 
called the second order of differences. 

3. Proceed, in like manner, for the thirds fourth^ Ji 
&c. orders of differences ; and so on till they termin 
or are carried as far as is thought necessary. 



EXA.MPLES. 

1. To find the several orders of differences iu the 
ries 1^ 4, 9, 16, 25, 36, &c. 

1, 4, 9, 16, 25, 36, Sec. 
3, 5, 7, 9, 11, &c. \st diff, 
2, 2, 2, 2, &c. 2flf diff, 
0, 0, 0, &c. 

2. To find the several orders of differences in the sei 
1,8,27,64, 125, 216, &c. 

1, 8, 27, 64, 125, 216, &c. 

7, 19, 37, 61, 91, &c. \stdiff. 

12, 18, 24, 30, &c.2rfflfe;^. 

6, 6, 6, &c. 3c/ diff^ 

0, 0, 5cc. 



•^I^ATION OF SERIES. 1 67 

To find the several orders of differences in the 
s 1, 3, 6, 10, 15, 21,&c. 

jin€. let 2, 3, 4, 5, is^c, 2d 1, 1, 1, IS^c. 

To find the B^eiral otders of di£Perences in the series 
,^0,50, 105,196, &c. 

^na/Ut 5, 12, 30, 45, 91, ^c. 2(/9, 16, 25, 36, 
ere. 3rf 7, 9, 1 1, ^c. 4th 3, 2, ^c. 

PROBLEM II. 

arieaj a, A, c, <f, f, ^c. d^m^* grven^ to find the first 
ttrm of the nth order of differences, 

RULE*. 

;t ^ stand for the first term of the nth differences. 

• . ^, .. w — 1 n — 1 n — 2. 

ben wHl a — nb+nx — -—c — nx ■ X — r-flf + 

— 1 n — 2 w — 3 „ . , t. , 
-- — X — ^ — X — — e, &c. to n + 1 termsss^, when 

an even number. 

id — fl 4- «3 — nx-— — c + wx— T— X — 5— c/ — n x 

- X s- X e, &c. to n + 1 terms = ^, when n 

3 4 

odd tia'mber. 



Vhen the several orders happen to be very- great, it will be 
converaeht to take the logarithms of the quantities conGemed, 
I differences will be smaller; and, when the operation is 
^ the quantity answering to the last logarithm may be easily 



168 



SUMMATION OF SERIES. 



EXAMPLES*. 



1 . Required the first term of the third order of diffe- 
rences of the series 1| 5, 15, 35| 70, &c. 

Let a, b^ c, d^ e^ life. = 1,5, 15, 35, 70, ^c. and n « 3. 

' Then — a + nb — n X c + » X --r— X — -— rf = 

2 2 3 

— a + 3d— 3c 4- d = — 1+15 — 45 + 35=4 = the Jirat 

term required. 

2. Required the first term of the fourth order of dif- 
ferences of the series 1, 8. 27, 64, 125, &c. 

Let a, d, c, d^ e^ ks^c. = 1, 8, 27, 64, 135, ^c. and 

na=4. 

^. ^ . n-i— 1 n— 1 n — 2. 

7%tfna — wd + nx— -— c — nx— r— X — -r—d+nX 

72 — 1 n — 2 w — 3 , . ^ 

— — X— T— X e=a — 4d+ 6c — 4rf+ e=l— -3^ 

+ 162 — 256-1- 125=0; ao that the Jirat term of the 
fourth order ia 0. 

3. Required the first term of the fifth order of diflRe- 



rences of the se<ies 1^ ^, } -^^, 8cc. 



4. Required the first term of the 8ih order of diffe- 
rences of the series 1, 3, 9, 27, 81, Sec. •^n«..256. 



* The labour in these kind of questions may be often afairidgedbf 
^tting cyphers f6r some of the terms at the beginnings ofui€ if* 
lies; by which means several of the differences will be equal tff^ 
and thie answer, on that account, obtained in fewer terms. 



SUMMATION OF SERIES. 169 



PROBLEM III. 
To find the nth term qfthe aeries Uy by r, d, e^ ^c. 



RULE. 

Let cfS rf ", flf"S d^y &c. be the first of the several 
orders of differences found as in the last problem : 

. 77—1 . . . 72—1 7Z— 2 , . 72—1 

Then wUl a+-r— fl^'+-r— X— r-^" +— T" X 
fi.^3 n — 3 ,„. . 77—1 71 — ^2 77 — 3 n — 4 , 

Ice. be asnth term required. 

EXAMPLES. 

1. To find the 12th ter^of the series, 2, 6, 12, 
20, 30, Sec. 

2, 6, 12, 20t 30, &c. 

4^ 6, 8, 10, &c. 

2, 2, 2, &c. 

0, 0, &c. 

Here 4 onrf 2 or^ the first terms qf the differences ; 
Lciy therefiire^ 4a(/>, 2&=flf'S and n:=s\2. 



Thena+'^d^+^^X^^dii^2+\\d' + 

55^"»2+44+lU)aBl56a 12rA /frui, or the anstver 
required, 

P 



170 SUMMATION OF SERIES. 

2. Required the 20th term of the series 1, 3, 6^ 10, 
J 5, 21, &c. 

1, 3, 6, 10, 15, 21, 8cc. 

2, 3, 4, 5,' 6, &c. 

1, 1, 1, 1, &c. 

0, 0, 0, &c. 

Here 2 and 1 are thejirat tenna of the differences. 
Let^ therefore^ 2=s:d^i !«=:</'", and n=s%Oy 

77-^-1 7?— —I 7^— — 2 

Th€na+ d^+ X-5-rf"=l + 19rf' + 

l71c?"=5l4-38+171=210=20/A rtrm required. 

3. Required the 15th term of the series 1, 4, 9, 16, 
25, 36, &c. ^ J^ns. 225. 

4. Required the 20th term of the series, 1, 8, 27, 64, 
125, &c. .4n8. 8000. 



PROBLEM IV. 



To find the sum of n terms of the series c, by r , d^ e^ Is^c. 



auLE. 



Let rfS rf", cf"S tf»^, &c. be the first of the several 
orders of differences. 



Then wiU «fl+nx^^d' + n x ^— X --"^^.x <^" 
w— 1 w — 2 w — 3 72 1 n 2 « 

+ „X ^ X — X — xd-+»x^x V^ 

72—3 72-— 4 , 

—J— X — j-«'^ &c = to the suqi of n terms of th« 
Befits, 



SUMMATION OF SERIES. 171 



SXAMP^ES* 

1. *To find the sum of n terms of the series 1, 2, 3, 
4> 5) 69 &c. 

1, 2, 3, 4, 5, 6, &c. 

1, 1, 1, I, 1, Sec. 

0, 0, 0, 0, See* 

Here 1 and are thejir%t ierma of the differences: 

Let J therefore^ a=s I, d^sss 1, and e/ii=0; 

_ / w — I, . «2 — n w2 + n 
7%^ lutllna + nx—T-'dissn'^ — = — - — =s sum 

qfn terma^ as required, 

2. To find the ^Su^-iSf»'«; terms of the series 1*, 2^, S^, 
4*, 52, &c. or 1, 4, Q^fl^, 2i5;'8cc. 

1, 4, 9, 16, 25, &c. 
3, 5, 7, 9, &c. 
'2j /2,'!;j2,"8cc. .. 

: .;d, 6, Sec,- •. . 

^(?re 3 and 2 ar^ the Jtf^t terms of the differences : 
Let^ therefore^ as=:l, di=3, awrf rfii=2. 

Then nvillnd +'« X -i^II-rfi+nx -^^ X — l-c/"=72H- 
• ' - '2 2 3 

^ , n — 1 . ^^ w.i— 1 71.^2 • 3n3 — n w^ — 3««+27i 

.^2^23 2:^ 2 

«X(«+l)x(2n4.l) ^ . - . , 

.«s i i—^ i SSI sum ofn terms J as required. 



* Any term of a given series, or the sum of any number of its 
terms, may be accuratelx determined, when the differences of any 
order become sftlasrequal to each other. 



4 

172 SUMMATION OF SERIES. 

3. To find the sum of n terms of the series P, 2^, 3», 
4^, 53, 8cc. or I, 8, 27, 64, 125, &c. 

1, 8, 2r, 64, 125, Sec. 

r, 19, 37, 61, &c, 

12, 18, 24, &c. 

6, 6, &c. 

0, Sec. 

Hence thejirat terms of the differences are 7, 12,awd6. 
Letf therr/orcy aal, d^osTt (/iial2, and d^^^m^e. 

2 2 3 

n— -1 w-^2 n — 3^,,, . ^ « — 1 , ,^ 

+ WX -r- X -T— X— 7— rf"i«»+7«X-T- +1^ 
« 3 4 J 

n — 1 n — 2 . n — 1 n — 2 72^—3 7n*— S» 

4 4 

=s 5MM ©/"n terms f as required. 

4. To find the sum of n terms of the series 2, 6, U 

20^ 30, 8cc. , «X(«+ l)XC«+2; 

3 

5. To find the sum of n terms of the series 1, 3, 6, IC 
15, See. ^ n n+l «+S 

1 2 3 

6. To find the sum of n terms of the series 1, 4, IC 
20, 35, Sec. . n Ti-fl n+2 «+S 

7. To find the sura of n terms of the series 1*, 2*, S^ 

43, Sec. or 1, 16, 81, 256) &c. 

. n» w4 «3 n 

5 ^ 2^3 5 



SUMMATION OF SERIES. 1 73 



PROBLEM V. 

TTie aeriea a, ^, c, «/, Cf i!fc, being given j whose terms 
^T€ an unit's distance jfroni teach other ^ tojindany interme- 
^atc term by interpolation, 

RULE. 

Let X ht the distance of any term y to be interpo- 
la ted, and d\ d", d"i, &c. the terms of the dif- 
ferences : 

Then will a+xe/i+arX^^d^+arX^^^X 

—dni+xy.^— X -^ X -j-rf-', &c. =y. 



EXAMPLES. 

1. Given the logarithmic sines of 1° o', 1° 1', 1° 2', 
and 1"* 3', to find the sine of 1° 1' 40". 

To' . l** I' 1*^2' 1°3' 

Sines 8.2418553 8.2490332 8.2560943 8.2630424 

71779 706 H , 69481 

—1168 — 1130 

38 
Here the first termsofthe differences are7l779 — 1 168, 
mnd 38. 

Lett thereforcy ar==l*' 1' 40"— 1° 0'«l'40"=r lf= 
distance ofyy the term to be inter/iolated ; and d^^sT 1779, 
<f Us 1168, a7i(i</"i=s38. 



_ ,f X— 1 

Then fw/^yaoa-fardi+^X— t-c^"+J?X— ^-X 

2 ^ 

'^'^i/"i-«a+f/i+4t/"— ^"i=»8.2418553+ 

P 2 



ir4 SUMMATION OF SERIES. 

.0119631— ..0000694 — .0000002 =8.25 375 33 s tine qf 
1* 1' 40", a9 was required. * 

2. Given the series ^ ^ ^^ ^ ^-j, &c. to find 
th« term which stands in the middle between i^» and ^. 

3. Given the natural tangents of SS** 54^ SS"* 55', 
88** 56', 88'* 57', 88** 58', 88** 59', to find the tangent 
of 88** 58' 11". Jina. 55.711144. 



PROBLEM IV. 

Having given a aetiea qf equidistant terms^ a, 6, r, d, f, 
l^c, whose Jir at differences are small ^ tojind any intermc' 
diate term by imterfiolation. 

RULE*. 

Find the value of the unknown quantity in the equa* 
tion which stands against the given number of terms, ia 
the following table, and it will give the term required. 

f . a — b^» 

2. 0^—3*+ c= 

3. a — Zb+ 3c — rf= 

.4. a— 4d+ 6c^ 4rf+ ^= 

5. a — 5.^+ 10c — 10flf+ Se — /=0 

6. 0.^66+ 15^—20£r+ 15c — 6/*+^«0 

n-.-! n— »1 rr i g 

7. a-^d+nx— r-c— -nx— -- X— r—e;^ b'c. «aO. 

4 3 3 



• The move tenos ive givea the mom accucatd/ the eqvatioa 
will sppraxhMsitv. 



SUMMATION OF SERIES. 175 



EXAMPLES* 

I. Given the logarithms of 101, 102, 104) and 105 ; 
to find the logarithm of 103. 

Here the number oftermBOte 4. 
Therefore against 4) in the tablej we have a— 4ft+6^-** 

id+essO; or c== — ^ — ^ 

o 

as 2.0043214 

Whence 2 *= 2.0086002 
irnence ^ ^^ 2.0170333 

<?» 2.0211893 



4X(d+</)sl6.1025340 
a+e SB 4.0255107 



6)12.0770233 



2.0128372a:/oj'. of 103, at re- 
quired, 

3. Given the cube roots of 45, 46, 47> 48, and 49 ; to 
find the cube root of 50. ^na. 3.684033. 

3. Given the logarithms of 50, 51, 52, 54, 55, and^56; 
to find the logarithm of 53. jim. 1.7242758695. 



PROSCXSCUOUS XXAMPLBS HBLATXVO TO SBBIES. 

1. To find the sum (^) of » terms of the series 1, 2, 3^ 
4, 5, 6, 8cc. 

Firtt^ l+2+3+44-5,Urr.....fi8B^. 
jtndn+ (n— l)+(n— 2)+(n— 3)+Ci>-^), tf'c 



ire SUMMATION OF SERIES. 

Therefore («+i)-f (n+l)+(n+l)+(n+l), ^c. 

And consequently (n+ l)xns=25,' or 5= =«ttOT 

required, ^ 

2* To find the sum (S)o^n tenns of the series 1> 3, 5, 
7, 9, 1 1, &c. . 

JF^rsty l-(-3+5+r+9,&c (2n— 1)=5. 

^nrf (2«— 1)+(2« — 3)+(2w— 5)+(272— r)+ 

(2n— 9), e5*c 1=5. 

There/ore 2»+2«+2«+2«+2n+, Isfc 2n^2S. 

Andj consequently^ 2nXnsgs2Sj or Sss — . — issn^ss 
sum required, "* 

3. Required the sum (5) of n terms of the series a+ 
\a+d)+la+2d)+{a+Sd)+{a+4,d)y &c. 

Firsts a+{a+d)+(a+2d).+(^a+3d)y ^c a+ 

Qi—l)Xd^S. 

And a +(nd — d)+a+(^nd — 2d)+a+(nd — 3d)+a+ 
Qnd—4.d)j ^c a=:5. _ 

Therefore 2a+(n(/— rf)4^i||fefoicr--^)+2fl+(m/ — d)-, 
i^c.2a+(nd—d)t=2S. ^ "^'W 

Andy consequently^ {2a+nd-^)Xn=s'2S; or 5=(2a-^ 

wc/;— ^)X —-=»««»» required. 

OR thus: •- . 

* > - ■ • 

Firsty a+{a+d)+(a+2d)+(a+Sd)+(a+4,d), ^c. 
^ $(+l + l + l + i + l, ^c.xa > o 

1(4-0+1+2^+3+4, ^c.XdS "^ 
JButn terms ^1+1+1 + 1 + 1, ^c.= «, 

Andn terms o/o+ 1+2+3+4, erc.«^2i!lZli2. 



SUMMATION OF SERIES. 177 

Therefore S:::xna + "^^'^'^^^^^» (2a + nd—d)x 

n 
— , aa before. 

4. To find the sum (S) of n terms of the series 1, x^ 

X^y J?3j ^4j gcc. 

Firaty 1 +a;+x«+a:3+xS tl^c x^^xbS. 

And x+x^+x^+x^-^-x^^ is^c x^^sSx, 

Tkerrfore — l+x^saiSx — S; 

Or Ssss srsaum required. 

Andj when x ia a firofier fraction^ the aum of the aeriea^ 
continued ad infinitum^ may befiund in the aame manner* 

Thuaj 1 +x+x^+x^+x^j ^c.bsS. 

Jnd X+X*+X^+X^+X'f IS^C.msiSx. 

Therefore — il ss^— ^/ or S^Sxbs 1. 

Whence^Sssi xmaum qfan it\fini$e number qfterma^ 

1— ^ar 

aa required. 

5. Required the sum (S) of the circulating decimal 
•999999, 8cc. conttnued ad infinitum. 

First, .999999, to-c. =^ + ^ + ^ + ^, l:fc. 

^ 10 ^ 100 ^ 1000 ^ 10000 

1 1 . 1 . 1 «- ^ 
10 100 ^ 1000 ^ 10000 9 

1 1 1 10.S 
Therefore X + r^ + r^^^^^^c.^^. 

T" sT 9 ' 



Therefore 5=< 



178 SUMMATION OF SERIES. 

6. Required the sum (5) of the series fl^+(«+^^i 
(a+2d)2+(a+3cf)2+(a+4c/)«, &c. continued to n tcrnT^ 

First ^ a2=flS 

(a+ d)2=a«+2Xlarf+ \d^ 
(a+2d)2=a2+2x2ad+ 4flf2 
(a+3^)2=a2+2x3acf+ 9rf« 
Ca+4rf)2=sa»+2X4acf+ 16d» 
e)^c. ere. 

'Sumxifn terms q/*(l + l + l + l,^c. 

Xa* 
+ . . . rfiW 0/ (0 + 1+2 + 3, ere. 

X2ac/ 
+ . . . cffV/o of (0+1+4+9, ere. 
L Xc/2 

Butn terms q/* 1 + 1+ 1 + 1, £5*c.==n. 

i>tV/o 0/0+1+2+3, erc.=!2fc:l), 
•^«rf (/z«o (/0+ 1+4+9, erc.^'i^ii!!!:!!^^":- 

^ * 1X2X3 

rr^^enci? ^=nxag+wxry2— l)xarf+''^^"~^^^^^""" ^^ 

1X2X3 

1X2 X3 
«wm required. 



7. Required the sum (5) of the series a»+(a+c/)^-f 
(a+2cf)3+(a+3c?)3+(a+4rf)3, &c. continued to n terms. 

First f a^ssa^ 

(a+ </)3=.a3+3Xla2c?+3X lflrf«+ Id^ 
(a+2d)33=a3+3x2a2rf+3X 4flrf2+ . 8rf3 
(a+3rf)3=a3+3x3a2</+3X 9acf2+ 27^3 
(a+4rf)3=a3+3X4c2cf+3Xl6a</s+ 64rf3 

(a+5d)3«a3+3X5o3rf+3X25fld2+135</3 
err. is^c* 



SUMMATION OF SERIES, 179 

SuTnofn$erm9o/{\ + l + l + Uisrc.) 

+ ... ditto ©/• (0+1+2+3, ere.) 

^ i<> J xSa^d 

ejore, o-<; ^ _ ^fg^ ^ (0+1+4+9, erSr.) 

Sad^ 
H ditto of (0+1+8+27, i!fc.) 

I terms o/^l + l + l + l, ^fr. ==» 

...0/0+1+2+3, ere, =r^^ig=:i^ 

. . . 0/0+1+4+9, ere. ^!!X(— i)x(2«-i) 



0/0+1+8+27+64+125, eTc. = 



1X3X3 

n3 — 2n3+«2 



2X3 



o « . «XC»— I)x3a2(/ . 

equently the sum S^rstiXarA ^ r- h 

^ ^ 1X2 

*\) X (2w~l) X 3g rf» (n^— 2n3 +n8) x d^ 

1X2X3 2X2 *" 

ruired** 

Required the sum (4^) of n terms of the serie$ 

r+15+31, &c. 

erma of this series are evidently equal to l^ •(*+2), 
-4), (1+2+4+8), ^c. or the successive sUms of 
^netrical firogression 1, 2, 4, 8, 16, eTc. 
therefore^ a== 1, ancf r=2, and we shall AflVtf a+ar 
ar^+ar^i eTc. =1 + 2+4+8+16, e^c. 



an account of figurate numbers, with the methods of find* 
sums, &c. I must refer the learner to Simpson'i Algebra, 
Behere he will find this subject fUly explained. 



1 80 SUMMATION OF SERIES. 

But the 9umoflf2y 3, 4, is'c. terms of tfda aeriea 

2. t -„ (r«— 1) X -2_ 



3. -;;:3p « (r^-1) X j£j 



*' -TUT ^ ^^'^ ^ 7=i 



•^ ' r I i — ntermMo/l + \ + \ + ljCrc^ 

But 1 + 1 + 1 + 1 + 1+1 + 1, e^c. c=«. 

Whence 5=(r* — 1)X — «)X L « ^unrequired. — 

r— 1 r— 1 

9. * Required the sum of («) terms of the scric^^ 
1.3 7 15 ,31 ^ 

7%tf terms of the series are the successive sums qfthegeo - — 

metrical progression ^+i+i+T+-A'> ^^* 

a a 
Lety therrforcy fls=l andrs^^y /^ff« tw^c+— +— + 

TT+7---, ^5*0. continued at pleasure. 
64 138 



Many qoestiimi of tUt kind, as wcU as several other thiqgi.ie. 
IsAtig to series in general, may he fonnd in Dodson^ MaflK natkM |v 
Repository, whkh is an txcd&ai analytical performance. 



•I 



SUMMATION OF SERIES. 181 

ut the auma of 1, 2, 3, 4, ^c. terma of (his series are 
(r — I) xa a 

(r3— 1) xa 1 a 

(r3_l ) XI , 1 a 



(r — 1) xr2 ^ 7-2' r — I 

(r— .1) x»3 ^ r3^ r— 1 
IsTc. ^c. 



Cntermaqfr +r'^-r'{-rjiS*c, 
r— I / — « terms o/-+ -;+--+ — cJJ'c. 



JJm^ r+r+r+r, ^c. s=;ir. 

1111 r« 1 



1 r r2 r3' (/ — l;x/""-^ 

a 7*»— . t 

IVhence S=s • x (nr — ; - )=a?/;;z rtt/u/rcd. 

r — 1 ^ (r — l)Xr"--* ' 

10. To find the sum (5) of the infinite series of the 

^ciprocals of the triangular numberS) — | f- 1 , 

<:c, 13 6 10 

Let-- — (-—-4.-— + --, Isfc. ad ivjinitiun = S, 
1 3 D 10 

Or, — + — + -i +J-, eJ'c ^ s. 

' 1.1 t.3^2 3 2.5 

Then -L + J- + J- + J-, l:fc = — . 

•1.2 "^ 2.3 ^3.4 4.5 2, 

'^'f-4)*(T-i)+(T4)+a-i). 



182 



SUMMATION OF SERIES. 



Ory 



11111 

— I 1 1 1 ^c. 

1^2^3^4^5 



S 1111 



J^=T 



5 1 

Whence y —-=—-; or 5=2= sum required, 

11. * To find the sum of n terms of the seru 
I 1 1 1 1 



1 



10 ' 15 



1 1 1 I 1 1 

12^345' » 

.r,, 11111 1 

Then z -= — + _ + f- _, isTc. tc—. 

1 2^345 n 



And r— • f 



1 



1 1111 

l'72+l 2^34^5 n+ 



• The figurate numbers, of which the terms of this and sevei 
other series are the reciprocals, may be exhibited thus : 

{1st. order") ri, 1, 1, 1, 1, &c. 

2d. order 1, 2, 3, 4, 5, &c. 

3d. order \ are J 1, 3, 6, 10, 15, &c. 

4th. order | 1, 4, 10, 20, 35, &c. 

5th. orderj U» .^» 15, 35, 70, &c. 

It may also be remarked, that different series are ^stinguish 
by particular names, according to the nature of their terms. 

Thas, a series, whose tenps continually decrease, is c^lled-a \ 
verging series. 

And a series, whose terms neither increase nor diminish, is caU 
a neutral series. 

Thus: 1— i+i-.i+|, &c. converging ; 1—2+3—4+5, fi 

diveigiDg: and 1— 1+1— l+i &c. neutral. 



SUMxMATION OF SERIES. 183 

therefore =- + -+— .-| .e^Tcro — . 

^ 1 72+1 2^6^12^20' n 72+ 1 

^ n 1.1.1 1 1 

Or = h -^, isTc, to . 

72+1 2^6^12^20' w.(/i+l) 

Tf-, 272 1 . 1 . 1 . 1 c^ 2 

iv hence = f- , ^c. to -i . 

72+1 1 3 6 ' 10 72.(n + l) 

a= —7— s=su7n of the eertesy or answer required. 

72+ 1 

12. Required the sum of the infinite series +- 

_L + -L. + J_, &c. 

<«*0«4 0»4«d 4*d*D - 

Let r= ■T- + "r + "Q"^T"*"T'' ^^' ^^ infinitum. 

Then z— — = ^'T + ■T"^ — » ^^* *^ trans/ioaition, 

^72d 1 = h ' h — 4" — 1 ^c. ^y subtraction 

1.2^2.3^3.4^ 4.5' ^ 

Or 1 — — = \ 1 1 , ^c. 6y transfiosition, 

2 2.3^3.4^4.5 5.6' ^ 

vf72cf — = 1 1 T— 9 ^c* by subtraction. 

2 1.4.3^2.9.4^3.16.5' 

1 2 , 2 . 2.; , 2 

2 1.2.3^2.3.4^3.4.5^4.5.6 

mence 1-^2=^ + 5^;^ + ^, Isf cad infinitum. 

^^, i. ^ 2= i.; therefore -1^ + ^ + ^, con^ 

tinned to infinity y is equal to — , nvhich is the sum required. 

Hi 



184 SUMMATION OF SERIES. 

1 



13. To find the sum of n terms of the series 



1.2.3 

I 1 I « 
H 1 , &c. 

2.3.4 3.4.5 4.5,6 

Let r= 1 1 (-, ^c. to — • — ; — -. 

1.2^2.3^.3.4^' w.(w+l) 

Then z-^ — sa 1 1 , Is^c, lo 



2 2.3 3.4 4.5 • n.(n+l) 

2 (n4-l).(72+2) 2.3^3.4 4.5^5.6 

^*"'«^'""' T- („+i)I(„+3) " rli + d:* + aXi' 

{contirtued to n terms) by subtraction. 

Whence — — — * » ■• — , 

4 2.(w + l).(»4-2) 1.23^ 2.3.4 3.4.5* 

i^c» (^continued to n terms)^ by division, 

*^ndi conaequentlyy + 1 , continued to 

ia^.3 2.3*4 3.4(5 

n terms, is^^ — — -— ; — : — ; ■ . ■ =: sum required. 
' 4 2.(n+\),{n+2) * 

14. Required the sum (S) of the series ■^— h -r 

1 
■-r» &c. continued ad infinitum, 

► Let x=s — ^ and S:=s 



2 ' 1+07 

z 
Then --^j — ssX'^^x^+x^^-^x^+x'j tJ'r. 
i-jrx 

.4ndz^{\+x)X{x — x2+x3 — x^, eJ'c.) 



SUMMATION OF SERIES. 185 

Whence^ by ^x — x^+x^'--~x^+x^^ ^c. 



:ey by C a 
icatioTiy ^ 1 



?7iultifilication^ C ^^"-^ 



X X^-\-X^'^X^'\-X^^ ^c. 

Whose sum is s= x+0+O+O+Oy ^c. 
Therefore z^=iX* 

X 

And JT— 072 + x^ — x^+x^^ ^c, = 



sum 



\+x' 
,11111 41 

2 4^8 16^32' 1+^ 3 

required* 

12 3 

15. Required the sum of the series f- f- h 

248 

4 . 5 



^ 



+ ^&C. 



2 



Let 07= -i-, awrf iS= 



2 ' (1— ar)a' 



7'^fn ri Y2=J^+2x2+3x3+4x4+5ar*, eJ'c. 



^«(/ z=(l— .r)2x(a:+2a:2+3j;3+4^4+5x5^ ^Tc.) 
Whence^ i 
fnulti/iticati 



JVhence, by ^x+2x^+3x^+4x^f ^c. 
i/iticaUonj ^ 1 — 2j: + x* 



x+2xi+3x^+4,x^, Ufa, 

— 2x^ — 4x3 — Qx^^ l!fc, 

+ x^+2x^,^c. 



Whose sum is ssx+O -fO + 0, ^c. 

There/ore x=sz, 

And J7+2x24-3x3+4x4+5x«, e^^c.! 



X 



(1— x)s* 
12 3 4 5 4 

2 ^ 4 ^ 8 ^ 16^32' n— 4)a 

required^ ^ 



186 SUMMATION OF SERIES. 

16. Required the 8um (S) of the infinite series 

, ere. 

3^^ 9 ^27^81 



Then '—^-—xax+4.x^+9x^+\6x^+2SxSy IsTc. 
( 1— x)3 

^nrfz=.(l— a;)3x(r+4x2+9x3+16x^ ISfc.) 

^=X'\-x^y as will be found by actual multijilication. 

There/ore x+x^xsz. 

^nd conaeguently x+^^+9x^+ 1 6x^, i:fc,^—r — "xT 



^ I . 4 . 9 . 16 ,^ 7X(l+4) 3 I 

3^9^27^81* (1— •J)3 2 2 

9Um required, 

17. Required the sum (S) of the infinite seues 
a a+d a+2d a+Zd 

m mr mr^ mr^ 

1 5' 

Let JT a ag ■■, owe/ 5 



4 w.(l — ^)2 

rrt^ f g , a+d ^ a+2d , a+3d ^^ 

m (1— x)2 m Twr mr^ mr^ 

^ r . n+rf . a+2d , a+3d ,_ 

(» — ^r 
{ii+3rf)XJ^', ^c. 

^nrf z=(l— a:)2x(a+Ca+r/)r+(a+2d>2+(a+3rf) 
x3),^c.s»(l-^)Xa+rfJ?>«« ««// afifiear by actual multi- 
filication. 

Therefore z^[\ — :p)Xo+c/x. 



SUMMATION OF SERIES. isr 

> , . a , d+d , a+2d . . 

jfnd consequent ly^ — }- — -| r-, cTc. = 

iw tnT iwr* 

(1 — ^x)Xfl+rfj: 

-:= «wm q/'Me injinite aeries required t 



EXAMPLES FOR PRACTICE. 

1. To find the sum of n terms of the series a+(a-^^) 
-f(a— 2cf)+(a— 3(/)+(a— 4rf), &c. 

Jna, — X(2a— 7J — IXrf). 

2. Required the sum of the infinite series a+rfa-f 
d^+d^a+d^a^ &c. where cf is a proper fraction. 

Ana. ■ 

1— rf 

S. Required the sum of the infinite series l + 3x+ 

6ar»+lOj;3+15x<, Etc. ^ I 



4. Required the sum of the infinite series l+4x+ 
I0x3+20:c3-^35^4j &c. . 1 



(I— ;c)4- 

5. Required the sum of the infinite series r-r+ — + 

l«o 3*5 

-+ ~, 8CC. ^«.. -. 

6. Required the sum of 40 terras of the series (1X2) 
+(3X4)+(5X6)+(6x7)+(rx8), &c. 

Ana, 22960. 

7. To find the sum of the infinite series 1+2^x4- 
34^2+44x3+54 1:4, 8cc. ^' l + llj?+Ux>+ll.r3 

Ana. , , ' ' \j — ■ ■. 

8. Required the sum of the infinite series -7-r-rr 
^ 2.3.4.5 S.4.5.6 ' ^»*. 



188 SUMMATION OF SERIES. i 

9. Required the sum of n terms of the series 
I 2.3.4. 5. 6„ 

-+ -+ T3+ 7J+ T5+ ^' ^'^' ^ 



r 7' 



^n«. — 



r—l r(r— I) 
10. Required the sum of n terms of the series 
I . 1 . 1 



f :rTT^+ TTT^> &c. 



1. 2.3.4 2.3,4.5 3.4.5.6 

. J ^ 

'**' 18 3.(«+ . {71 + 2) . (»+ ^> 

1 1 \ 

11. Required the sum of the series— -+ —- + -^r; 

1.4 A,5 —3.0 

^ 4J* ^""^ nX(3+w)- 

11 . n , n TE. 

18* 3+37i 12+6« 17+ ^n 

12. Required the sum of the series —- ^ -f 

2.6 4.8 

1 c ^ ^ ^ 2 o .5n+3«* 

-, &c. r-7 — ■ . vfw«. 2= — , o=- 



6.10' 2«(44-27i) 16' 32+48»4-iew2 

1 1 

13. Required the sum of the series h 

^ 4.8 6.10 

1 ^ 1 

8.12 ' • (2-f-2w).(6+2«)* 



48 16-f-16« 36 + 24« 

14. Required the sum of the series 1 1* 

3*8 6.12 

I . 1 i 



9.16 12.20' 3w.(4+4w) 



^729. Sssr , iSsss— .. 



12 12+12» 



SUMMATION OF SERIES. 189 

15. * Required the sum of the infinite series 1 

+ — , &c. 

3 4 5 6 

jina, .69314718, ^c. or hyfi, log. of2» 

I 
" 16. Required the sum of the infinite series 1— -— + 

, &c. 

5 7 9 11 

jina. .78539, &c. or \ cir.ofthe circle whose diam, is 1. 

1 2 

17. Required the sum of the diverging series — 

3 4 5 2 3 

-I — — I , &c. w^«*. .193 147, ISTc, or l+hy/u log. of 

4 5 6 a 1 

18. Required the sum of the diverging series 
22 32 42 52 

5 

Jns. 1.943147, 8cc. or 1- hyfi. log, of 2. 

4 

19. tRequired the sum of the hyper-geometrical series 
1—14.2 — 6+24—120, 8cc. or 1 — 1a+2b — 3c + 4d 

— -5e, Sec. ina, near afifir ox. -value ss.298174. 

* A great variety of series, of different forms, may be found in 
other authors ; but those which are here given ^vUl be sulEcient for 
the learner's practice. 

The names of the principal authors, who have written upon this 
subject, are as follows : 

Archimedes ; Arabes ; D* Alembert ; Barrow ; Briggs ; Nicholas, 
Daniel, John, and James Bernoulli : Fermat; Descartes; Clairaut; 
C<mdorcet{ Cotes; Dodson; Euler; Emerson; Fagnanus; Le 
Grange; Goldbach ; Gregory; Halley; Harriot^ Huddens; Huy- 
gens; Button; Kepler; Keil ; Landen; Madaurin; DeLagney; 
Leibnitz ; Lorgna ; Lucas de Burgo ; Manfredi ; Monmort ; De 
Moivre t Montano ; Nichole ; Newton ; Oughtred ; Riccati ; Reg- 
nald; Saundersoni Sterling; Slusius; Simpson; Brook Taylor; 
Varignon; Vieta; WalUs; Waring; &c. 

t For an account of these series, with a new method of finding 
thcdr i^iproximate values, see Hutton^s Mathematical Tracts, 
lately published. 



[ 190 ] 



OF LOGARITHMS^. 



Logarithms are numbers so contrived and adapted tea 
other numbers, that the sums and differences of the 
former shall correspond to, and show, the products and 
quotients of the latter. 

Or, more generally, logarithms are the numerical 
exponents of ratios ; or a series of numbers in arithme- 
tical progression, answering to another series of numbers 
in geometrical progression. 

7y 5^' ^' ^' ^' ^' ^' Indices or logarithms. 
' ^ 1, 2, 4, 8, 16, 32, Geometric firogression. 

Or 5^' ^' ^' ^' ^' ^' Indices or logarithms, 
' C 1> 3, 9, 27, 81, 243, Geometric progression. 

Or 5^' ^' ^» ^» ^' ^' Ind.orlog. 

' C 1> 10, 100, 1000, 10000, 100000, Geo. prog. 

Where it is evident that the same indices serve= 
equally for any geometric series ; and consequently^ 
there may be an endless variety of systems of logarithms^ 
10 the same common numbers, by only changing tht^ 
second term, 2, 3, or 10, &c. of the geometrical series 



* The invention of logarithms is the undoubted right of Lortf 
Napier, Baron of Merchiston, in Scotland, and is properly considered 
as one of the most useful and excellent discoveries of modem times. 
A table of these numbers was first published by him at Edinbuighf 
ann. 1614, in a treatise entitled Canon Mirificum Logarithmonakf and, 
as their great utility and extensive application were sufficiendy appa. 
rent, they were immediately received by all the learned throughout 
Europe, Mr. ffenry BriggSt Saviiian professor of Geometry at Chcfordf 



OF LOGARITHMS. 191 

It is also apparent, from the nature of these series, 
^t, if any two indices be added together, their sum 
'^l be the index of that number which is equal to the 
Pfoduct of the two terms, in the geometric progression, 
'0 which those indices belong. 

Thusy the indices 2 and 3, being added together^ are = 
^ i and the numbers 4 and 8, or the terms corresponding 
^ith those indices^ being multifilied together^ are = 32, 
^/dch is the number answering to the index 5. 

And, in like manner, if any one index be subtracted 
K*om another, the difference will be the index of that 
^ umber which is equal to the quotient of the two terms 
^ which those indices belong* 

Thua^ the index 6, minus the index 4, {> e= 2 ; and the 
ierma corres/ionding to those indices are 64 and I6y whose 
quotient is =: 4> ; which is the number answering (o the 
index 2, 



upon hearing of the discovery, set out upon a visit to the noble in- 
ventor, and soon afterwards they jointly undertook the arduous task 
of computing new tables upon this subject, and reducing them to a 
more convenient form than that which was at first thought of. But 
lord Napier dying before they were Hnished, the whole burden fell 
upon Mr. Briggs^ who, with prodigious labour and great skill, made 
an entire Canon, according to the new form, for all numbers from 1 
to 20000, and from 9000 to 101000, to 14 places of figures, and 
published it at London, in the year 1624, in a treatise entitled yfriV/;. 
metica Logarithmica, with directions for supplying the intermediate 
ckiiiads. 

This Canon was again published in Holland by Adrian Vlacq, aifho 
1628, together with the logarithms of all tlie numbers which Mr. 
Bn'ggs has omitted ; but he continued them only to 10 places of 
decimals. Mr. Briggt also computed the logarithms of the sines, 
'tangents, and secants to every degree, and _ i^ part of a degree of 
the whole quadrant, and subjoined them to the natural sines, tan. 
gents, and aecants, which he had befoie computed to 15 places of 



192 OF LOGARITHMS. 

For the same reason, if the logarithm of any number 
be multiplied by the index of its power, the product will 
be equal to the logarithm of that power. 

Thusy the index or logarithm q/*4, in the above aerte»f 
ia 2 ; and^ if this number be multifilied by S, the firoduct 
wil 6r = 6; which ia the logarithm of 64j or the third 
flower of 4. 

And, if the logarithm of any number be divided by the 
index of its root, the quotient will be equal to the loga- 
rithm of that root. 

Thua, the index or logarithm of 64 ia 6 ; andj if thia 
number be divided by 2, the quotient will 6^ = 3; which 
ia the logarithm ofS^ or the aguare root 0/64, 

The logarithms most convenient for practice are 
such as are adapted to a geometric senes increasing 



figures. And these tables, together with their construction and use, 
were first published in the year 1633, after Mr. £riggs*s death, by 
Mr. Henry Gellibrand, under the title of TVigoiwmetria Britamuea. 

Benjamin Ui sinus has also given us a table of logarichms to every 
10 seconds. And Mr. Wolf^ in his Mathematical Lexicon^ sayv that 
one Van Loser had computed them to every single second, but his 
untimely death prevented tneir publication. 

A great numbet of otlur authors have treated on this subject, but 
as their numbers arc freqaently inuccurate and incommodiously dis- 
posed, they are now ger>eral7y iieglecied. The tables in most repute 
at present axe those of Gardiner, in 4to. first published in the year 
1742, and Sherwin's tables. In 8vc. first printed in the year 1705, 
where the logarithms of all numbers may be easily found from 1 to 
1000000 ; and those of the sines, tangents, and secants, to any de- 
gree of accuracy required. 

Dodson's Jntilogarithmic Canon is likewise a veiy ingenious workt 
being of ^reat use for finding the numbers answering to any given 
logarithms. 

Smoe the first publication of this wodk, Mr. Miehad Taylor** tables 
have appNeared, coniainmg the common logarithms, and the loga- 
jSthmic unes and tangents to every second of the quadrant. 



OF LOGARITHMS. 193 

in a tenfold proportion, as in the last of the above forms ; 
and are those which are to be found, at present, in most of 
the common tables upon this subject. 

The distinguishing mark of this system of logarithms 
is, that the index, or logarithm, of 1 is ; that of 10, 1 ; 
that of 100, 2 ; that of 1000, 3, &c. And in decimals the 
logarithm of 1 is — 1 ; that of .01, — 2 ; that of .001, — 
3; &c. 

From Whence it follows that the logarithm of any num- 
ber between 1 and 10 must be and some fractional partSi 
and that of a number between 10 and 100 will be 1 and 
some fractional parts; and so on for any other number 
whatever. 

And since the integral part of a logarithm is always 
thus readily finind, it is usually called the index, or charac* 
teriatic ; and is commonly omitted in the tables ; lieing 
left to be supplied by the operator himself, as occasion re* 
quires. 



OF THE „ 

MAKING OF LOGARITHMS. 

Whatever arithmetical progression we apply to a geo* 
metrical one, the terms of it are logarithms only to that 
leries to which we apply them, and answer the end pro* 
posed only for those particular numbers ; so that if we had 
logarithms adapted only to particular geometrical seriesi 
they would be bat of little use. The great end and de- 
sign of these numbers is the ease and expedition which 
they afibrd in long calculatbns, by saving the laborious 
work of mtilii/Uicafioni diviaion^ and the extraction of 
r%ot% i but this end would never be completely answereli 

R 



194 OF LOGARlTHxMS. 

unless logarithms could be adapted to the whole system 
of numberst 1)3,3, 4, &c. And as here lies the chief excel- 
lence and merit of the contrivance, so also the difficulty. 
For the natural system of numbers, ly 2, 3, 4, &c* being 
an arithmetical, and not a geometrical series, seems rather 
fit to be made logarithms of, than to have logarithms ap* 
plied to it. But this difTiculty may be easily removed, by 
considering. 

That though the whole system of natural numbers, I, 

3, 3, 4, &c. is not in geometrical progression, and cannot, 
by any means, be made to agree with such a series, yet 
they may be brought so near it, as to be within any assign- 
able degree of approximation ; which may be conceivtrd, 
in general thus : supixise a fraction indefinitely small to be 
represented by J7,atid a geometrical series arising from 1, 
in the ratio of 1 to 1 +^> to be 1, (1 + x)*, (1 + jr)2, 
( 1 + ^)^>( 1 + ^)^j &c. Then must some of these tenns 
come indefinitely near to all the natural numbers, 1, 2,3, 

4, &c. ; because, amongst numbers which arise by ex-^ 
tremely small increments, some of them must exceed, or^ 
fall short, of any determinate number, by an mdefiniteljr^ 
little excess or defect.. 

If, therefore, in the places of the terms of this series^ 
which approach indefinitely near to any of the natural 
numbers, we suppose these natural numbers themselves 
to be substituted, then will this series be in geometrical 
progression, to an exactness which may be caUed^'ncf^ntf r; 
because the approximation of its terms to the natuni 
numbers can never end, but goes on in infiniium. 

And since this imagined geometric series-comprehendi^ 
xkidefinitely near, the whole system of natofalnilmblrrs; Iv 
3*^ 3, 4) &c. so the indices of its terms comprehend a- 
Vhfile system of logarithmsi which arer adapted- to this 



OF LOGARITHMS. 195 

system of numbers, and may be extended to any length 
we please. For though the natu^i system of numbers 
make not, by themselves, a complete j^eometrical series, 
yet they are conceived as a part of such a series, and their 
lopjarithms arc tlie indices of their distances from upity in 
that series ; or, more generally, they are the correspond- 
ing terms of an arithmetical series applied to that geome- 
trical one. 

But, again, it must be observed, that an indefinitely 
small fraction cannot he assigned ; and« therefore, in the 
actual construction of logarithms, we must be contented 
with a determinate degree of approximation. Whence, 
according as we take x^ in the series, 1, ( I + J^)S (1 +J^)*> 
(1 + jr)3, (1 + xy, &c. the approximation of its terms to 
the natural numbers will be in different degrees of exact- 
ness: for the lessor is^ the nearer will be the approxima- 
tion ; but then the more are the number of involutions of 
1 + ^> necessary to come within any determinate degree 
of nearness to the natural number assigned. 

Thus then we uaay conceive the ix)S8ibility of making 
logarithms to all the natural numbers^ 1, 2, 3, 4, &.c. to any 
determinate degree of exactness; viz. by assigning a very 
: small fraction for jt, and actually raising a series, in the 
ratio of 1 to 1+ ^9 iind taking for the natural numbers 
such terms of that series as are nearest to them, and their 
indices for the logarithms. Hut then, to construct loga- 
riihms in this manner, to suqh an extent of numbers, and 
degree of exactness, as would be necessary to make them 
of any considerable use, is next to impossible, because of 
the almost infinite labour and time ii would require. This, 
however, is an introduction for understanding the method 
of the nolfle inventory who undoubtedly first took the hint 



196 OF LOGARITHMS. 

of making logarithms from the consideration of the in- 
dices of a geometrical series; and by means of the prin- 
ciples and known properties of these progressions he first 
formed his tables, and adapted them to the practical pur- 
poses intended. 



PROBLEM L 

To find the logarithm of any of the natural numbers^ 1, 2, 
3, 4, ^c, according to the method of Napier. ; 



RULE*. 

1. Take the geometrical series, 1, 10, 100, 1000, 10000, 
Sec. and apply to it the arithmetical series 1, 2, 3, 4, &c. as 
logarithms. 

2. Find a geometric mean between 1 and 10, 10 and 
100, or any other two adjacent terms of the series betwixt 
which the number proposed lies. 

3. Between the mean, thus found, and the nearest ex- 
treme, find another geometrical mean, in the same man- 
ner ; and so on, till you are arrived within the proposed li- 
mit of the number whose logarithm is sought. 

4. Find as many arithmetical means, in the same order 
as you found the geometrical ones, and the last of these 
will be the logarithm answering to the number required. 

* The reader who wishes to inform himself more particularly 
concerning the history, nature, and construction of Ic^nthroi, may 
consult Hutton's Mathematical Tables, lately published, where he 
will find his curiosity amply gratified. 






OF LOGARITHMS. 197 



EXAMPLES* 

Let it be required- to find the logarithm of 9. . 
Here the numbers between which 9 lies are 1 and 10. 

J^'irsty then, the log, af\0 ia L, and the log, qf\ t« 0; 

1 + 
ther^ore - " w. ^» ss .5 i% the arithmetical mean, and 

y/{\ X IO)=v^lO=3.l622r77 = 5'co»jtfrricmea«; whence 
thv logarithm of S A 622 777 is .5. 

Secondly y the log. of \0 is 1, and the log, of 

1 4. 5 
3.1622777 is .5 ; therefore — =s .75 a= arithmetical 

jnea7iyand^{\0 X 3.1622777) = 5 6234132 =s geometric 
iHciin: whence the log. q/*5.6234132 is ,75. 

Thirdlyy the log, of \0 is 1, and the log, (/ 5.6234 132 

1+75 
fs .75-f t her (fore — ■ = .875 =as arithmetical mcan^ 

<and 5/(H).X5.623413:;i) s 7 A9S942\ss geometric mean : 
whence the log. (/ 7.498942 1 is i875. 

Fourthly, the log. of Wis l, and the log. of 7.498942 1 

is .875 ; therefore — r-^ « .9 $7 S ^^ arithmetical mean, 

and ^( 10 X 7.4989421) SB 8.6596i43 1 «»/8r«ro»iff/nc mean: 
whence the log. qftj659^3\ is .9375. 

Fijthly, the log. of 10 i* I, and the log, of 8.6596431 

1 -^ 93T5 
is ;93r5 ; .thert^fort ^^^ le .96875 « arithmetical 

mean, and •(lO X 8.6596431) =9.3057204 »:5'Mm(?m'c 
mean: whence tht log. of 9-3057204 is .96875. 

'R2^ 

/ 

/ c» 



19S OF LOGARITHMS. 

Sixthly^ the log. 0/ 8.6596431 is .^STS, and the log, of 

.9375 4- 96875 
9.3057204 w .96875 ; therefore ^ =.953125 

=flWrA.mean,andv^(8.6596431x9.3057204)=8.9768713 
^=t geometric mean: whence the log, qf 8 .97687 13 is 
.953125. 

jind^tjiroc ceding in this manner^ after 25 extractions^ 

the logarithm o/" 8.9999998 Vfill be found to be .9542425 ; 

which may be taken for the logarithm of% because it dif 

fers from it only by 505Jofl6 > and is therefore sufficiently 

exact for all firacticatfiurfioses. 

jind in the same manner the logarithms of almost all the 
prime numbers were found i a work so incredibly laborious^ 
that the unremitted industry of several years was scarcely 
sufficient for its accomfilishment, 

PROBLEM IL 

To determine the hyfierbolic logarithm (l) (f any give?i 

number Csy, s - ■>. 

The hyperbolic logarithm pf any number fs)the^dex) 
of that'term of the logarithmic progression which agfees 
with the proposed number multiplied by the excess of the 
common ratio above unity. 

* Let, therefore, (I + x)« be that term of the logarith- 
mical progression, 1,(1 + x)\ ( 1 + x^^ ( 1 + x)», ( I + j:)S 
Sec. which is equal to the required number (n). 

Then wiU (I + jc)« = n, and 1 + a? = «»; and if 

I + y be put as N, and m =58 — ^ wc shall have 1 -f x 

= N« == (1 + y)»» 1 + my + m X^i^ya + m K^Ii^ 



OF LOGARITHMS- 199 

m — 1 m — 2 

And, consequently, x =fny+mx — —y^+^nX—^ — 

X y\ &c. where m being rejected in the factors 

Tn-I- 1, »2 — 2, m — 3, &c. being indefinitely small in com- 
parison of I, 2, 3, &c. the equation will become ;cs=7/iy — 

2^3-4' 

Hence _ (n:r = i.)= y — 1- +' ^ — L. + ^ &c.= 
m 2 3 4 5 

hyperbolic logarithm of n, as was required. 



PROBLEM IIL 

The hijfierbolic logarithm (l) of a number being gtven^ to 
Jind the number (n) itself^ which answers to it. 

Let (14-^)" be that term of the logarithmic progres- 
sion 1, (1 + ^)S (1 +^)S i^+^Yi (1 + J^)S &c. which 
is equal to the required number n. 

Then, because ( 1 + x)" is universally = 1 + war + n x 

^ — i « . w — 1 w — 2 , ^ .... 

— r — x2+7zx — - — X — - — J^^i occ. we shall have I + nx 

n — 1 n — 1 n — 2 

+ nx -T--^^ + nx — T— X — T— -^^ ^c» = N. 

But since tz, from the nature of the logarithms, is here 
t supposed indefinitely great, it is evident that the numbers 
^( ^' connected to it by the sine — may all be rejected, as fiir 
^ as any assigned number of terms* 

For as 1, 3, 3, &c. are indefinitely small m compariflion 
to n, the rejecting of those numbers can iPcry litCle affect 
the values to which tliey betoog • 



» 



200 OF LOGARITHMS. 

If; therefore, 1, 2, 3, Sec. be thrown out of the factors 
— r — I — - — , , &c. we shall have 1 + nx -J 

+ "TTTT + ;: J &c.= N. 

2.3 2.J.4 

But nx (= l) is the hyperbolic logarithm of (1 + :r)«, 
or K, by what has been before specified ; and therelbre 1 

1% l3 1.4 

+ L + — H + ~ — » &c.=s N = number required. 

2 2,3 2<3.4 



PROBLEM IV. 

To determine the hyfierboUc logarithm (l) of any j*iven 
number (n) by a universally convergiiig aeritu, 

t/2 y3 yA 

The series y — ~ + -^ , &c. is the most easy 

and natural that can be obtained ; but, in determininp^ the 
logarithms of large numbers, it is but of little use, since, 
in such cases, it diverges instead of converging. 

Let, therefore, the number whose logarithm you would 

find be denoted by , and also let (1 + •^)" be the 

i — y . 

term of the logarithmic progression agreeing with the 

proposed number. 

Then (I + x)n = -L-; or 1 + ^ = _L^x «, 

(I— y)-^ = (l— y)"*(by putting m== )a»l— my+ 

n 
w— I m — 1 m-— 2 „ ^ 

m X — T— y».-^m X — T— X — r— y3,»kc. 



OF LOGARITHMS. 201 

Whence y + -- H 1 , &c. = = nxasz hy- 

^2^34' m ^ 

perbolic logarithm of ; which series, it is manifest, 

\ — y 

will constantly converge, let the value of be ever 

so great ; because y will always be less than unity. 

But it is to be observed, that this series, except 
in its signs, has exactly the same form with that 
above found for the logarithm of 1 + y, and that, 
if both of them be added together, the series 3y + 

9y3 2t/* 2r/^ . . 

— ^ H — '• — h -~-, Sec. thence arising, will be more simple 

than either of them^ as one half of the terms will, in that 
case, be eniirely destroyed. 

Since, therefore, the sum of the logarithms of any 
two numlws is equal 'to the logarithm of the pro- 
duct of those numbers, it is manifest that 2x -f 
2^3 2j7* 1 -4- X 
—^ + ■— , Sec. will truly express the logarithm of ; 

which series converges still faster than jc -] 1 , Sec. 

not only because the even powers are here destroyed, but 
because x, in finding the logarithm of any given number 
(n), will have a less value. 

And, in order to determine what this value of x must 

\ -{- X N— — I 

be, make , = n, and then x will be found = — r-^: 

1 X N 4" 1 

p 
but if the quantity praposed — be a fraction, instead of a 

whole number, make =» — > 

1 _j7 q 



202 OF LOGARITHMS. 

P *— Q 

and we shall have 3 = i; and either of these 

values being substituted in the foregoing series 2jc + 

r- — I > &c. will give the hyperbolic logarithm of the 

number required. 

Now, by finding JVa/ner'a logarithm of any number, 
according to the foregoing method, Biigga^Si or the com- 
mon logarithm of the same number, may be found as 
follows : 

Briggs's logarithm of any number is to J\/a/iter*8 loga- 
rithm of the same number, as liriggs'a logarithm of 10 
is to J\ra/tier*8 logarithm of 10. 

But Brigga*s logarithm of 10 is 1, and JVa/iier's loga- 
rithm of 10 is 2,302585093; whence, if Brigga's^or the 
common logarithm of any number, be denoted by c. l. 
and JVafiier^Sy or the hyperbolic logarithm of the same 
number, by h. l. we shall have 2.302585093 : 1 : : 

I 
H. L. : c. L,; or h. l. X = H. L. X 

2.O02585093 

.4342944819 =« c. l. as was required*. 



* There are, besides these, many other ingenious methods, which 
later writers have discovered, for tiiidmg the logaiithms of numbers 
in a much easier way than the original inventor; but, as they can- 
not be understood without a knowledge of some of the higher 
branches of the mathematics, 1 have thought proper to omit them, 
and must beg le ve to refer the reader to those works which are 
written expressly upon the subject. 

It would likewise much exceed the limits of this compendium, to 
point out all the peculiar artifices that are made use of for construct, 
ing an entire table of these numbers ; such as thofe of Gardimr, 
Skerviin^ and others, who have treated on this subject ; but any in- 
formation of this kind, which the learner may wish to obtain, may 
be found in JButton'a Tables, before mentioned. 



OF LOGARITHMS. 



A TABLE OF LOGARITHMS. 

Havinf^ explaineil the mcihotl of making a table of 
the logariifims of numbers greaiir than imity, ibc next 
thing to be done is lo show how the logantlims of frac- 
tional quantities may be found. And, in order lo Ihisi 
it may be observed, that, ai we hjve hitherto supposed a 
^omclric series to increase from a unit on the right 
hand, to we may now suppose il to decrease from a unit 
towards the left; and (he indices, in this cascj being 
made nei;ittive> will still exhibit the logarithms of the 
terms to which ihcy belong. 

r/ms-T-of. — .1— 2 — 1 0+ I + a + 3, ts'r. 
^"'"■TWTil tV ' '" 10O lOOO.iS'c. 
Whence -\- 1 ieihf t'.gaH'hm of\0,and — 1 the logarithm 
c/^j ; +11 lite logartl/im of IQO, and — 2 the losarilhm 4 

And from hence it appears, that all numbers, consisting 
of the same figures, whether they l« integral, fracttonaU 
or mixed, will have the decimal parts of their logarithms 



will be sunicici 



fm 






ghid, X 



wof .1 



foond only by mfxns of jddiiion and it 

3t l lO-=t-B-)-i.. 5; L,3 = L 10 — t_3i L. 6i=L.3+i_ 3i u>d 

10 on for uiy olhcr of tboe numlien. 

In like niuiHOr the Ion of a X A ^ i- <■ + I" ^1 <h* leg. of 
s-^A^L. a — L. A| th«log. of r" sBHi^r, and ilial ih« bg. of 



204 OF LOGARITHMS. 

Thu9ythe logarithm of 587^ being 3.7689339, the loga- 
rithm of^ jjp T^cm ^^' fi^^^ ^f^^ ^^ *^ asfollofoa : 

Num. Logarithms* 



5 8 7 4 

5 8 7.4 

5 8.7 4 

5.8 7 4 

.5874 

.05874 

.005874 



3.7 6 8 9 3 3 9 

2.7 6 8 9 3 3 9 

1.7 6 8 9 3 3 9 

07689339 

— i.7 6 8 9 3 3 9 

— 3.7 6 8 9 3 3 9 

—3,7 6 8 9 3 3^9 



From this it also appears, that the indea: or charac- 
teri9tic of any logarithm is always one less than the num- 
ber of figures which the natural number consists of; and 
this index is constantly to be placed on the left hand of 
the decimal p9rt of the logarithm. 

When there are integers in the given number, the in- 
dex is always affirmative ; but when there are no inte- 
gers, the index is negative, and is to be marked by a line 
drawn before it, like a negative quantity in algebra. 

Thus J a number having 1, 2, 3, 4, 5j ^c. integer 
filacesj 

The index of its log, is 0, 1,2, 3, 4, isfc. 

jlnd a fraction having a digit in thefilace of firimes^ ae* 
condsy thirds, fourths^ ^c. 

The index qfits logarithm will de •— 1, — 2, — 3| — 4, 

It may also be observed, that, though the indices of 
fractional quantities are negative, yet the decimal parts 
of their logarithms are always affirmative ; and all opera- 
tions are to be performed by them in the same manner'as 
by negative and affirmative quantities in algebra. 






\. 



OF LOGARITHMS. 205 

In taking out of a table the logarithm of any number, 
not exceeding 100000, we have the decimal part by in- 
spection ; and if to this the proper characteristic be af- 
fixed, it will give the complete logarithm required. 

But if the number, whose logarithm is required, be 
above 100000, then find the logarithm of the two nearest 
numbers to it that can be found in the table, and say, as 
their difference : the difference of their 1 )garithm8 : : 
the difference of the nearest number and that whose 
logarithm is required : ibe difference of their logarithms, 
nearly} and this difference being added to, or subtracted 
from, the nearest logarithm, according as it is greatek* 
or less than the required one^ will give the logarithm 
required, fiearly. 

Thus, let it be required to find the logarithm of 
367182*. 

The decimal fiart o/S67\ i«, by the tablcy 5647844 \and 
0/3672 ia .5649027: 

.-. The 5 367100 is 5.5647844 > 
log. of I 367200 ia 5.56490-27 J 

Their diff. 100 0001 183 diff. 

JSTeareat JSTo, C 367200 
Given .Vo. ^367183 

18 diff. 

*■ This method, ^ing founded on the supposition that the loiga* 
rithm9 of all numbers between 367100 and 367200, increase or de. 
crease, equally, according to their distance from 367100 or 367200, 
is not strictly tfue, but nearly so; and the greater any numbers are, 
«ith respect to their difference, the nearer will those differences be 
proportional. And, therefore, though this method will not give the 
exact logarithm, yet it wlU be a very near approjutnation, aad is 
su^ciently exact for most practical puipop- 

b 



206 OF LOGARITHMS. 

Therffore 100 : .0001183 : : 18 : .0000212. 

w^nrf 5.5649027— .0000212 = 5.5648815 = logarithm 
of ^67 \^% nearly . 

If the number consists both of integers and fractions, 
or is entirely fractional, find the decimal part of the loga- 
rithm as if all its figures were integral ; and this, being 
prefixed to the proper characteristic, will give the loga- 
rithm required. 

And if the given number is a proper fraction, subtract 
the logarithm of the denominator from the logarithm of 
the numerator, and the remainder will be the logarithm 
sought; which, being that of a decimal fraction, must 
always have a negative index. 

And, if it is a mixed number, reduce it to an improper 
fraction, and find the difference of the logarithms of the 
numerator and denominator, in the same manner as be- 
fore. 

In finding the number answering to any given loga- 
rithm, the index, if affirmative^ will always show how 
many integral places the required number consists of: 
and, if negative^ in what place of decimals the first, or 
significant figure, stands; so that, if the logarithm can 
be found in the table, the number answering to it will al- 
ways be had by inspection. 

But, if the logarithm cannot be exactly found in the 
table, find the next greater, and the next less, and then 
say. As the diff*erence of these two logarithms : the 
difference of the numbers answering to them : : the 
difference of the given logarithm and the nearest tabular 
logarithm : a fourth number; which added to, or sub- 
tracted from, the natural number answering to the near- 
est tabular logarithm, according as that logarithm is less 
or greater than the given one, will give the number re- 
quired, nearly. 

Thus, let it be required to find the natural number an- 
swer)' oganthm 5.5648815. 



OF LOGARITHMS, S07 

The next less and greater logarithms^ in the table^ are 
5.5647844 > The numbers C 367 100 
5.5649027 3 answering ^367200 

Th eir diff. .0001183 1 00 

And 5.5649027 — 5.56488 15=.00002 12. 
Therefore .0001183 : 100 : : .0000212 : 18 nearly. 
IVhence 367200 — 18=377182= WMmd^r required^. 



MULTIPLICATION BY LOGARITHMS. 



BULK. 

Add the logarithms of the factors together, and their 
sum will be the logarithm of the product required. 

Observing to add what is to be carried from the de- 
cimal part of the logarithm to the sum of the affirmative 

indices : 

And that the difference between the affirmative and ne- 
gative indices is to be taken for the index to the logarithm 
of the product. 



* Directions, at large, for the using of logarithms, may be found 
in most of the common tables upon tliis subject. Shenoin'n Mathe- 
matical Tables, of the edition 1741 or 1742« are reckoned the most 
correct and convenient, for practk:al purposes, of any now extant, 
except those of Dr. Hutton, lately published; which, besides their 
accuracy, are much better arranged; and, in the two first degrees, 
the sines. &c. are given to every second He has also a table ot hy- 
perbolic logarithms, and several others equally usefol. 



208 OF LOGARITHMS. 



EXAMPLES. 



1. Let the number 256 be multiplied by 4. 
The log. 0/256 = 2.4082400 
The log. of 4 = 0.6020600 



Thefiroduct = 1024.... 3.0)03000 

2. Let the number 8.5 be multiplied by 10. 
The log- o/S.5 =0.9294189 
The log. of 10^ 1. 0000000 



The firoduct = 85 .... 1 .9294 189 

3. Let the number 46.75 be multiplied by .3275. 
The log. 0/46.75 = 1.6697816 
The log. o/'.3275= — 1.5152113 



The firoduct =± 15.31 ....1.1849925 

4. Multiply 3.768, 2.053, and .007693 continually to- 
gether. 

The log. of 3.768 = 0.576 1 109 
The log. qf 2.053 = 0.3 123889 
The log. 0/ .007693 = — 3.8860997 



Thefiroduct^ .059511... — 2.7745955 

5. Multiply .5, .4, and .13 continually together. 
The log. of .5 =—1.6989700 
The log. of .4 = — 1.6020600 
The log. of.\2^ — 1.0791812 



Thefiroduct =.024.. 2.3802112 



OF LOGARITHMS. 2©9 



DIVISION BY LOGARITHMS. 



RULE. 

From the logarithm of the dividend subtract the loga* 
rithm of the divisor, and the number agreeing to the re- 
mainder will be the quotient required. 

But observe to change the index of the divisor from ne- 
gative to affirmative, or from affirmative to negative, and 
tlren the diffisrence of the affirmative indices must be taken 
for the index to the logarithm of the quotient. 

And, also, when a unit is borrowed, in the left hand 
l)]ace of the decimal part of the logarithm, add it to the 
index of the divisor; but, if it be negative, subtract it: 
ai!d let the index ^^^^"8^ ^''O"^ thence be changed and 
worked with as before. 

EXAMPLES. 

1. Let the number 56 be divided by the number ♦. 
The log. of 56 =s 1.7481880 
The log. of 4 = 0.6020600 



The quotient as 14 . . . 1.1461280 
3. Let the number 50.75 be divided by the numbei* 






The log. of SO J 5 = 1.7054360 

The log. of .25 = = — 1.3979400 



The quotient b 203 ...#••.. 2.3074960 



aid OP LOGARITHMS. 

3. Let the number .24 be divided by the number SO. 
The log. of.24i^ — 1.380^1 12 
nc log. ^ 80 a 1 .9050900 



The quotient .003 .... — 3.4/71212 

4. Let the number .01265 be divided by the number 
.35. 

The log. 0/ .01265 =« — 2.1020905 
The log. of .35 sss — 1.7403627 



The quotient = .023... 2.3617278 



INVOLUTION BY LOGARITHMS. 

RULE*. 

1. Seek the logarithm of the given number in the 
table. Jl 

2* Multiply the logarithm, thus ^und, by the index of 
the proposed power. 

3. Find the number corresponding to the product, and 
it will be the power required. 

JSTcte. In multiplying a logarithm with a negative in- 
dex, by any affirmative number, the product will always 
' be negative: 

But what is to be carried &om the decimal part of the 
logarithm will always be affirmative: 

And therefore their difference will be the index of the 
product; and is constantly to be made of the same kinA 
with the greater. 

* The rule of proportion is performed by addmg the Ipgaiathiiv 
of the last-two terms, and subtncdng the h^garithm of the first: 



OF LOGARITHMS. 211 



XXAMPLES. 



4. Required the second power of the number 3.874. 

The log. of 3.874 a= 0,5881596 

The index » 2 



The fiovfer « 15.01 .... 1.1763192 , 

2k Required the third power of the number 2.768. 

The log. of 2.768 = 0.442 1 66 1 

The index = 3 



The flower =a 5l,2l .. .. 1.3264983 

3*. Required the third power of the number .79 1 6. 

The log. of .79 1 6 »— 1 .8985058 
The index a 3 



7'he flower ^ .4961 .. ..—1.6955174 

4. Required the twel&h power of the number-1.539. 

The log. of 1.539—0.1872386 

The index mB 12 



The power » 176.6 .... 2.2468632 

5. Required the 20th power of 1.05. 

wfnt. 2.6533, ISTc. 

6. Required the 100th power of 1,05. 

Am. 131.50, e^r. 



212 OF LOGARITHMS. 



EVOLUTION BY LOGARITHMS. 



RULE. 

k Seek the logarithm of the given number in th 
table. 

2. Divide the logarithm, thus found, by the denomina- 
tor of the index of the root proposed. 

5. Find the number corresponding to this quotient, and 
it will be the root required. 

Xute, — When the index of the logarithm, to be divid- 
ed, is negative, and does not exactly contain the divisor, 
increase it by such a number as will make it exactly 
divisible, and carry the units borrowed, as so many tens, 
lb the left-hand place of the decimal, and then divide 
as in whole numbers. 

EXAMPLES* 

1 . Required the square root of the number 225. 

The log. of 225 = 2.352 1 825 

Therefore 2)2.352 1 825 



The root= 15 1.1760912 

2. Required the square root of the number 1 50 1 , 

The log. of 1501 = 3.1763807 

Therefore 2)3.1763807 



The root = 38.74 1.588 1 903 



MISCELLANEOUS QUESTIONS. 213 

3. What is the cube root of the number .16631^5 ? 

The log. of . 1 66375 = — 1 .22 1088 1 
T/iert/orr 3) — 1 .22 1088 1 



T/ie root = ,55 .... — 1 .7403527 

4. What is the square root of the number .08 1 62 ? 

The log. of .08162 = — 2.9117966 
Therefore 2) — 2.9U7966 

The root =s ,2857 ... — 1 .4558983 

5. What is the twelfth root of the number 176.6 ? 

The log. of 176.6 = 2.2469907 

Therefore 12)2.2469907 



The root » 1.539 1872492 



MISCELLANEOUS QUESTIONS. 

1. A person being asked what o'clock it was, an- 
swered, that it was between 8 and 9, and that the hour 
and minute hands were exactly together ; what was the 
time? h. ^ ^, 

Jim. 8 : 43 : 38y\. 

2. Divide the number 50 into two such parts, that \ of 
one part, added to ^ of the other, may make 40. 

.,ins. CO and 30. 

3. What two numbers are those, whose difference is 12; 
and their squares equal to each other? 

jina. + 6 and — . 6. 

4. There is a certain number, consisting of two places, 
which is equal to the difference of the Sf|uarc8 of its* di- 
gits ; and if 36 be added to it, the digits will be inverted ; 
quxre the number ? Jnti. 4a. 



214 MISCELLANEOUS QUESTIONS. 

5. Given x^ + ySacSl, and y^ ■{- x^ = 17 ; to find x 
and y, ^na x = 3 and y = 2. 

6. Given p' — xy = 666, and x^ + jt-i/ = 406 ; to find 
X and y, jins. x = 7 ««£/ i/ = 9. 

7. Given the sum of three numbtrs, in harmonical pro- 
portion, = 26, and tlieir continued product = 576 ; to find 
the numbers. jln^. 12, S^and 6. 

8. What two numbers are those, whose diff'crtncc, 
sum, and product are to each oilier as the numbers 2, 3, 
and 5, respectively ? ^fis,2a?id 10. 

9. *To find that number whose cube being subtracted 
from its square, shall leave the greatest remainder possi- 
ble. ^«*. |. 

10. It is required to find the least 3 whole numbers, so 
that ^ of the first, -^ of the second, and j^ of the third, 
shall be all equal to each other, jina, 28(j, 294, and 300. 

1 1 . Given zx^ + xz^ a» 290, and jc^ + 2^ = 64 1 ; to 
find X and z. ^na. x = 5, and z ss 2. 

12. Given the sum of three numbers in continued geo- 
metrical progression = 39, and the sum of their squares 
= 819; to find the numbers. *^na. 3, 9, 27. 

13. Required the least number of weights, and the 
weight of each, that will weigh from one pound to 29 
hundredweight, jina. 1, 3,9,27, 81, 243,729,fl«</2187. 

14. Required two numbers such, that their sums shall 
be equal both to their product and the difference of their 
squares. ^na. 2.618034 and 1.618034. 

1 5. It is required to find the least 4 affirmative integers 
such, that the square of the greatest may be equal to the 
sum of the squares of the other three, ^/^na, 3, 4, 1 2, and 1 3 . 



* This is properly a question in fluxions, but it is answered alge* 
braically by Mr. Emerson, as well as several others of the same 
nature. 



MISCELLANEOUS QUESTIONS. 215 

1 6. If money be lent, at three per cent. 

To those who choose lo borrow, 
In what time shall I be worth a pound, 
If I lend a crown to-morrow I 

.4718, 46.90036 yearsy allowing com/i, int^ 

17. Required the two least nonquadrate numbers, x and 
t/, such, that x2 + 1/2 and x^ + 1/3 shall be both square 
numbers. Ana, x = 364, and y = 273. 

18. There are three numbers in geometrical proportion 
such, that, if (he mean be subtracted from the sum of the 
two extremes, the remainder* muhiplied by the sum of the 
said two extremes will be 9^ ; but, if that remainder be 
multiplied by the sum of all the three numbers, the pro- 
duct will be 133 ; it is required to find the three numbers 
by a simple equation. jint. 4, 6, and 9. 

1 9. To determine two numbers whose sum shall be a 
cube, but their product and quotients squares. 

jlna, 4 and 14, 100 and 25, 900 and 100. 

20. Required that arithmetical progression whose num- 
ber of terms is 10, sum of the terms 185, and the sum of 
the cubes of the terms 104525. 

Jins. 5,8, n, 14, 17, 20, 23, 26,29, 32. 

21. To divide a given number (n) into 4 such parts, 
that if any other number («) be added to the first part, de- 
ducted from the second, multiplied by the third, and the 
fourth part divided by it, the sum, difference, product, and 
quotient, shall be all equal to each other. 

yn nn n nh x n 

22. Given ar^y + t/^^- = 512500, and x^y — y^xtss 
5500 ; to find xand y, Jina. xa3B23 and yss20. 

23. Given x + ?/ + z = 6,xy + xr + yz = 11, and 
xyz^6i to find JT, I/, and r. j4na» xas 3, y=l,fl72c/2=2. 



216 MISCELLANEOUS QUESTIONS. 

24. To find two numbers in the ratio of 5 to 7, M^hicli, 
btfing respectively divided by 9 and 13, shall leave 3 and 8 
for remainders. Ana. 2\Qy and 294. 

,25. To find three numbers such, that | the first, \ the 
second, and \ of the third, shall be = 62 ; l of the fii-stj-J 
of the second, and •} of the third «= 47 ; and \ of the first, 
\ of the second, and -J of the third = 38. 

Ana, 24, 60, a;2e/120. 

26. A, B, and C, are to share 100,000 pounds between 
them, in the proportion of ^, ^J, and -J, respectively ; but C's 
part being lost by death, it is required to divide the whole 
sum properly between the other two. 

Ana. Wafiareia 57142||f/. and B's 42857/^/. 

27. To find fdur numbers, z^yyZy and w, having the pro- 
duct of every three given; viz. Jcyz = 231, jryw = 420, 
yzw = 1540, and a:zw = 660. 

Ana, JT 3= 3, y = 7, z = 1 1, and w = 20. 

28. To find four numbers in geometrical proportioD, 
whose sum is 15, and the sum of their squares 85. 

Ana. 1,2,4,8. 

29. To find three numbers, x, y, and z, when the pro- 
ducts of each by the sum of the other two are given ; viz. 
^ X (y + z) = 48, y X (x -f z) = 39, and z x (x + y) 
= 63. Ang. J? = 4, y = 3, and z = 9. 

30. What number is that, which, being any how divid- 
ed, the square of one part, when added to the other part, 
$hall always be a square number ? Ana. 1 oniy. 

31. Given y3^2--i27,y3-f j;=l35, andx^ + ys ^ z^ 
fml\33i to find ;r, y, and z. Ana. x=:\0yyssSfandzss:2. 

.'32. Given jc^-f jcy=108,y2+ yz=69, and z^+ xz^ 
MO ; to find x, y, and z. Ana. xbs9, y=3, and z^=^0. 

S3t To find two mean proportionals between any two 
given numbers a and b. Ana. ^a^b and ^b^a. 



■/ 



MISCELLANEOUS QUESTIONS. 217 

34. Given \r+yz=s384, y+xz=23r, and z+:cy=s 
192 ; to find a:, y, z. Ana. x=:10, y=17, awrf z=22. 

35. To find the least number, which, being divided 
by 6, 5, 4, 3, and 2, shall leave the remainders 5, 4^ 3^ 
2, and 1, respectively. Ans. 59. 

36. To find three numbers such, that the sum or di£- 
fi^rence of any two of them shall be square numbers. 

Ana, 1873432, 2399057, ancf 2286168. 

37. To find two square numbers such, that their sum 
may be a square, and their difference a cube, and the 
side of the said square and cube equal to each other. 

38. To determine the number of fifteens that can be 
jiiade out of a common pack of 52 cards. Ana. 17264. 

39. Given the rates a and b of two ingredients, and 
(he rate c of the compound m, to find what portionaf x 
and y of each must be taken to compose the mixture. 

Ana. xsamx r, yssmX-^ — r 

40. Given x2+j:y+i/»=1087, and x*+x^y^+y^:=^ 
4577295 ; to find x and y» Ana, x=z2\ and yssl7. 

41. Given x+y+z^7Sj jr«+ya+z»=2546, and 
xy — j:z— yz&a527 ; to find Xj y, and z. 

Ana, xm:4i\j yssz2Syandz^9, 

s .1 

42. Given a:+ys=5 152, and (x — y^X^x — t^)Y=8192 ;: 
to find X and y. Ana. a:sl08, and yss44^ 

43. To find three numbers such, that if to the square 
of each the product of the other two be added, the sums 
<«hall be squares. Ana, 73, 9, 329* 

44. Let the number of cards in a pack (/;)• be 
distributed into any number of heaps (w), by laying 
OS many cards upon the bottom heap as arc sufficienlt 

T 



^■■ 



2 1 8 MISCELLANEOUS QUESTIONS. 

to make up its number (9) ; then, by having the number 
of cards remaining in the dealer's hand (r) and the num* 
ber of heaps (n) given, it is required to find the sum of 
all the bottom cards. 

Ana, (q+\)xn'\-(r'^)s=i autnreqtured. 

V 

45. To find 3 numbers such, that if each be subtracted 
from the cube of their sum, the remainders shall be 
cubes. Ana. fJlf^, ifffi, and ^^fj. 

46. Given the cycle of the sun 18, the golden number 
8, and the Roman indiction 10 ; to find the year. 

Ana* 1717. 

47. To find 3 cube numbers such, that their sum 
shall be both a square and a cube number : and if that 
aitfn be squared it shall be a cube, and if it be cubed it 
jfcdl be a squai-e. 4^^ ^ t, ^ * ^25 . 

8 '37^'216 

48. To find 3 numbers such, that if each be added 
to the cube of their sum, their sums shall be cubes. 

•^"*- TTTjiTT* rrnn> ttttst* 

49. With guineas and moidores, the fewest, which way 
Three hundred and fifty-one pounds can I pay? 

If paid every way 'twill admit of, what sum 

Do the pieces amount to ?— my fortune's to come. 

jina. 9 guineaa^ and 253 moidorea; and 37 waysf 
which ia ss 12987/. 

1 . 2 

50. Given x2/2*^=:z*^=slOO; to find the values of :f 

and y. Ana, x=:47.706 and zssslAi* 

51. Given x^'— 21j?2*+147jc*=316, to find the va- 
lue ofx. Ana. :rd 



r' - ■■'■ 



«* 



MISCELLANEOUS QUESTIONS. 219 

52. Given 44000xs+ 1 ss z^ ; to find x and z ia whole 

numbers. 

jins. xss:^04S29Sl22l7S\y and z^S49l78\\A%00\. 

53« To find three whole numbers suchi that the ex- 
cess of the greatest above the middle number shall be to 
the excess of the middle number above the leasts as 3 to 
I : and also that the sum of every two of these shall be 
squares. jina. 4nX 1362, 4nX402, and 4nx82. 

54. Given x+y^a (2), and j:«+y«=sA (32), to find x 
and y by quadratics. 

jins. aal.4697175, ancTy^.S 302824« 

55. Given xy=500, and y*=s300; to find x and y. 

^n«.xsa4.6914, and y=s5,Sl02. 

56. Given xyx(j:+z)*=300,aprx(y+z)a= 1296, and 
t/rx(jr+y)««432 ; to find a:, y, and z. 

wfn«. jrss], y=3| z^9. 

57. Given ii;3+j:+y+z = 57, w+j73+y+2-ai763, 
iif+x+y^+z^\:i50y and 5b;4- J^4-!/4-z3-b153; to find 
'^) Vi ^j and ti^. 

^TM. XKs 14| y=s 1 1| zss5y and w^S. 

58. Given jr + y=« 1750, J7z+yvs= 22708, xv+yz^^ 
12292, and xzv+vzysx 159252; to find x, y, z, and v. 

^na* x:b1743, y=s7| z^lS, and vmt7. 

59. Given 5x+7y+9z»93256; to find all the difie- 
rent solutions in affirmative integers which the equation 
will admit of. jina. 13801148. 

60. To find a square number such, that the sum of all 
its aliquot parts shall be a square number. Jina. 9401. 

61. To find two sc^uare numbers such, that either of 
them, when added to its aliquot parts, shall make the 
same sum. ^na. 106276 and 16S6A9. 



220 MISCELLANEOUS QUESTIONS. 

63. To find 4 whole numbers such, that the difference 
4)i every two shall be a square number. 

* ^72*. 1873432, 2288168, 2399057, and 6560657. 

63. To find three numbers such, that if their sum be 
multiplied by the first, it shall be a triangular number, 
by the second a square, and by the third a cube. 

64. To find three biquadrate numbers, the sum of 
which shall be a square. >4n8, 12^, 15S and 20^^. 

65. To find a right-angled triangle such) that its peri- 
meter sliall be a cube, and tlie perimeter, together with 
the area, a square. 

jfna. Per]i.^^i^\ *Wf=V^'» >iy/'-=\VA'. 

66. To find two different isosceles triangles su«h, that 
f-h^ir areas and perimeters shall be both equal. 

jitifi. Sides of the 072c=29, 29, 40. 

Ditto of the orAer=37, 37, 24, 
I* 

67. There is an island 73' miles in circumference, and 
lliree fpotmen all start together to travel the same way 
about it: A goes 5 miles a day, B 8, and C 10: when 
\rfll they all come together again ? ^ns, 73 days^ 

68. How much foreign brandy at 6s. fier gallon, and 
British spirits at 3s. fier gallon, must be mixed together, 
so that, in selling the compound at 9s. fier gallon, tlTb 
distiller may clear SO fier cent,? 

Ans, 51 gallons of brandy y and 14 of s/iii-iCfi^ 



THE END. 



"i 



-A^ 



i^l^^^^^BH 


THB NBW YORK PUBLIC LIBRARY 1 

REFERENCE DEPARTMENT H 


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